src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author hoelzl
Tue Mar 05 15:43:15 2013 +0100 (2013-03-05)
changeset 51344 b3d246c92dfb
parent 51343 b61b32f62c78
child 51345 e7dd577cb053
permissions -rw-r--r--
continuity of pair operations
     1 (*  title:      HOL/Library/Topology_Euclidian_Space.thy
     2     Author:     Amine Chaieb, University of Cambridge
     3     Author:     Robert Himmelmann, TU Muenchen
     4     Author:     Brian Huffman, Portland State University
     5 *)
     6 
     7 header {* Elementary topology in Euclidean space. *}
     8 
     9 theory Topology_Euclidean_Space
    10 imports
    11   Complex_Main
    12   "~~/src/HOL/Library/Countable_Set"
    13   "~~/src/HOL/Library/Glbs"
    14   "~~/src/HOL/Library/FuncSet"
    15   Linear_Algebra
    16   Norm_Arith
    17 begin
    18 
    19 lemma dist_0_norm:
    20   fixes x :: "'a::real_normed_vector"
    21   shows "dist 0 x = norm x"
    22 unfolding dist_norm by simp
    23 
    24 lemma dist_double: "dist x y < d / 2 \<Longrightarrow> dist x z < d / 2 \<Longrightarrow> dist y z < d"
    25   using dist_triangle[of y z x] by (simp add: dist_commute)
    26 
    27 (* LEGACY *)
    28 lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s o r) ----> l"
    29   by (rule LIMSEQ_subseq_LIMSEQ)
    30 
    31 lemmas real_isGlb_unique = isGlb_unique[where 'a=real]
    32 
    33 lemma countable_PiE: 
    34   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"
    35   by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
    36 
    37 subsection {* Topological Basis *}
    38 
    39 context topological_space
    40 begin
    41 
    42 definition "topological_basis B =
    43   ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x)))"
    44 
    45 lemma topological_basis:
    46   "topological_basis B = (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
    47   unfolding topological_basis_def
    48   apply safe
    49      apply fastforce
    50     apply fastforce
    51    apply (erule_tac x="x" in allE)
    52    apply simp
    53    apply (rule_tac x="{x}" in exI)
    54   apply auto
    55   done
    56 
    57 lemma topological_basis_iff:
    58   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
    59   shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
    60     (is "_ \<longleftrightarrow> ?rhs")
    61 proof safe
    62   fix O' and x::'a
    63   assume H: "topological_basis B" "open O'" "x \<in> O'"
    64   hence "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
    65   then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
    66   thus "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
    67 next
    68   assume H: ?rhs
    69   show "topological_basis B" using assms unfolding topological_basis_def
    70   proof safe
    71     fix O'::"'a set" assume "open O'"
    72     with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
    73       by (force intro: bchoice simp: Bex_def)
    74     thus "\<exists>B'\<subseteq>B. \<Union>B' = O'"
    75       by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
    76   qed
    77 qed
    78 
    79 lemma topological_basisI:
    80   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
    81   assumes "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
    82   shows "topological_basis B"
    83   using assms by (subst topological_basis_iff) auto
    84 
    85 lemma topological_basisE:
    86   fixes O'
    87   assumes "topological_basis B"
    88   assumes "open O'"
    89   assumes "x \<in> O'"
    90   obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
    91 proof atomize_elim
    92   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'" by (simp add: topological_basis_def)
    93   with topological_basis_iff assms
    94   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'" using assms by (simp add: Bex_def)
    95 qed
    96 
    97 lemma topological_basis_open:
    98   assumes "topological_basis B"
    99   assumes "X \<in> B"
   100   shows "open X"
   101   using assms
   102   by (simp add: topological_basis_def)
   103 
   104 lemma topological_basis_imp_subbasis:
   105   assumes B: "topological_basis B" shows "open = generate_topology B"
   106 proof (intro ext iffI)
   107   fix S :: "'a set" assume "open S"
   108   with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
   109     unfolding topological_basis_def by blast
   110   then show "generate_topology B S"
   111     by (auto intro: generate_topology.intros dest: topological_basis_open)
   112 next
   113   fix S :: "'a set" assume "generate_topology B S" then show "open S"
   114     by induct (auto dest: topological_basis_open[OF B])
   115 qed
   116 
   117 lemma basis_dense:
   118   fixes B::"'a set set" and f::"'a set \<Rightarrow> 'a"
   119   assumes "topological_basis B"
   120   assumes choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
   121   shows "(\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X))"
   122 proof (intro allI impI)
   123   fix X::"'a set" assume "open X" "X \<noteq> {}"
   124   from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X \<noteq> {}`]]
   125   guess B' . note B' = this
   126   thus "\<exists>B'\<in>B. f B' \<in> X" by (auto intro!: choosefrom_basis)
   127 qed
   128 
   129 end
   130 
   131 lemma topological_basis_prod:
   132   assumes A: "topological_basis A" and B: "topological_basis B"
   133   shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
   134   unfolding topological_basis_def
   135 proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
   136   fix S :: "('a \<times> 'b) set" assume "open S"
   137   then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
   138   proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
   139     fix x y assume "(x, y) \<in> S"
   140     from open_prod_elim[OF `open S` this]
   141     obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
   142       by (metis mem_Sigma_iff)
   143     moreover from topological_basisE[OF A a] guess A0 .
   144     moreover from topological_basisE[OF B b] guess B0 .
   145     ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
   146       by (intro UN_I[of "(A0, B0)"]) auto
   147   qed auto
   148 qed (metis A B topological_basis_open open_Times)
   149 
   150 subsection {* Countable Basis *}
   151 
   152 locale countable_basis =
   153   fixes B::"'a::topological_space set set"
   154   assumes is_basis: "topological_basis B"
   155   assumes countable_basis: "countable B"
   156 begin
   157 
   158 lemma open_countable_basis_ex:
   159   assumes "open X"
   160   shows "\<exists>B' \<subseteq> B. X = Union B'"
   161   using assms countable_basis is_basis unfolding topological_basis_def by blast
   162 
   163 lemma open_countable_basisE:
   164   assumes "open X"
   165   obtains B' where "B' \<subseteq> B" "X = Union B'"
   166   using assms open_countable_basis_ex by (atomize_elim) simp
   167 
   168 lemma countable_dense_exists:
   169   shows "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
   170 proof -
   171   let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
   172   have "countable (?f ` B)" using countable_basis by simp
   173   with basis_dense[OF is_basis, of ?f] show ?thesis
   174     by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
   175 qed
   176 
   177 lemma countable_dense_setE:
   178   obtains D :: "'a set"
   179   where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
   180   using countable_dense_exists by blast
   181 
   182 end
   183 
   184 class first_countable_topology = topological_space +
   185   assumes first_countable_basis:
   186     "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
   187 
   188 lemma (in first_countable_topology) countable_basis_at_decseq:
   189   obtains A :: "nat \<Rightarrow> 'a set" where
   190     "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
   191     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
   192 proof atomize_elim
   193   from first_countable_basis[of x] obtain A
   194     where "countable A"
   195     and nhds: "\<And>a. a \<in> A \<Longrightarrow> open a" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a"
   196     and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"  by auto
   197   then have "A \<noteq> {}" by auto
   198   with `countable A` have r: "A = range (from_nat_into A)" by auto
   199   def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. from_nat_into A i"
   200   show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
   201       (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
   202   proof (safe intro!: exI[of _ F])
   203     fix i
   204     show "open (F i)" using nhds(1) r by (auto simp: F_def intro!: open_INT)
   205     show "x \<in> F i" using nhds(2) r by (auto simp: F_def)
   206   next
   207     fix S assume "open S" "x \<in> S"
   208     from incl[OF this] obtain i where "F i \<subseteq> S"
   209       by (subst (asm) r) (auto simp: F_def)
   210     moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
   211       by (auto simp: F_def)
   212     ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
   213       by (auto simp: eventually_sequentially)
   214   qed
   215 qed
   216 
   217 lemma (in first_countable_topology) first_countable_basisE:
   218   obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
   219     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
   220   using first_countable_basis[of x]
   221   by atomize_elim auto
   222 
   223 lemma (in first_countable_topology) first_countable_basis_Int_stableE:
   224   obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
   225     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
   226     "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A"
   227 proof atomize_elim
   228   from first_countable_basisE[of x] guess A' . note A' = this
   229   def A \<equiv> "(\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)"
   230   thus "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and>
   231         (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)"
   232   proof (safe intro!: exI[where x=A])
   233     show "countable A" unfolding A_def by (intro countable_image countable_Collect_finite)
   234     fix a assume "a \<in> A"
   235     thus "x \<in> a" "open a" using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
   236   next
   237     let ?int = "\<lambda>N. \<Inter>from_nat_into A' ` N"
   238     fix a b assume "a \<in> A" "b \<in> A"
   239     then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)" by (auto simp: A_def)
   240     thus "a \<inter> b \<in> A" by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])
   241   next
   242     fix S assume "open S" "x \<in> S" then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast
   243     thus "\<exists>a\<in>A. a \<subseteq> S" using a A'
   244       by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])
   245   qed
   246 qed
   247 
   248 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
   249 proof
   250   fix x :: "'a \<times> 'b"
   251   from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this
   252   from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this
   253   show "\<exists>A::('a\<times>'b) set set. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
   254   proof (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
   255     fix a b assume x: "a \<in> A" "b \<in> B"
   256     with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"
   257       unfolding mem_Times_iff by (auto intro: open_Times)
   258   next
   259     fix S assume "open S" "x \<in> S"
   260     from open_prod_elim[OF this] guess a' b' .
   261     moreover with A(4)[of a'] B(4)[of b']
   262     obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'" by auto
   263     ultimately show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"
   264       by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
   265   qed (simp add: A B)
   266 qed
   267 
   268 instance metric_space \<subseteq> first_countable_topology
   269 proof
   270   fix x :: 'a
   271   show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
   272   proof (intro exI, safe)
   273     fix S assume "open S" "x \<in> S"
   274     then obtain r where "0 < r" "{y. dist x y < r} \<subseteq> S"
   275       by (auto simp: open_dist dist_commute subset_eq)
   276     moreover from reals_Archimedean[OF `0 < r`] guess n ..
   277     moreover
   278     then have "{y. dist x y < inverse (Suc n)} \<subseteq> {y. dist x y < r}"
   279       by (auto simp: inverse_eq_divide)
   280     ultimately show "\<exists>a\<in>range (\<lambda>n. {y. dist x y < inverse (Suc n)}). a \<subseteq> S"
   281       by auto
   282   qed (auto intro: open_ball)
   283 qed
   284 
   285 class second_countable_topology = topological_space +
   286   assumes ex_countable_subbasis: "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
   287 begin
   288 
   289 lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
   290 proof -
   291   from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B" by blast
   292   let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
   293 
   294   show ?thesis
   295   proof (intro exI conjI)
   296     show "countable ?B"
   297       by (intro countable_image countable_Collect_finite_subset B)
   298     { fix S assume "open S"
   299       then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
   300         unfolding B
   301       proof induct
   302         case UNIV show ?case by (intro exI[of _ "{{}}"]) simp
   303       next
   304         case (Int a b)
   305         then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
   306           and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
   307           by blast
   308         show ?case
   309           unfolding x y Int_UN_distrib2
   310           by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
   311       next
   312         case (UN K)
   313         then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
   314         then guess k unfolding bchoice_iff ..
   315         then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
   316           by (intro exI[of _ "UNION K k"]) auto
   317       next
   318         case (Basis S) then show ?case
   319           by (intro exI[of _ "{{S}}"]) auto
   320       qed
   321       then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
   322         unfolding subset_image_iff by blast }
   323     then show "topological_basis ?B"
   324       unfolding topological_space_class.topological_basis_def
   325       by (safe intro!: topological_space_class.open_Inter) 
   326          (simp_all add: B generate_topology.Basis subset_eq)
   327   qed
   328 qed
   329 
   330 end
   331 
   332 sublocale second_countable_topology <
   333   countable_basis "SOME B. countable B \<and> topological_basis B"
   334   using someI_ex[OF ex_countable_basis]
   335   by unfold_locales safe
   336 
   337 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
   338 proof
   339   obtain A :: "'a set set" where "countable A" "topological_basis A"
   340     using ex_countable_basis by auto
   341   moreover
   342   obtain B :: "'b set set" where "countable B" "topological_basis B"
   343     using ex_countable_basis by auto
   344   ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
   345     by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
   346       topological_basis_imp_subbasis)
   347 qed
   348 
   349 instance second_countable_topology \<subseteq> first_countable_topology
   350 proof
   351   fix x :: 'a
   352   def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"
   353   then have B: "countable B" "topological_basis B"
   354     using countable_basis is_basis
   355     by (auto simp: countable_basis is_basis)
   356   then show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
   357     by (intro exI[of _ "{b\<in>B. x \<in> b}"])
   358        (fastforce simp: topological_space_class.topological_basis_def)
   359 qed
   360 
   361 subsection {* Polish spaces *}
   362 
   363 text {* Textbooks define Polish spaces as completely metrizable.
   364   We assume the topology to be complete for a given metric. *}
   365 
   366 class polish_space = complete_space + second_countable_topology
   367 
   368 subsection {* General notion of a topology as a value *}
   369 
   370 definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
   371 typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
   372   morphisms "openin" "topology"
   373   unfolding istopology_def by blast
   374 
   375 lemma istopology_open_in[intro]: "istopology(openin U)"
   376   using openin[of U] by blast
   377 
   378 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
   379   using topology_inverse[unfolded mem_Collect_eq] .
   380 
   381 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
   382   using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
   383 
   384 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
   385 proof-
   386   { assume "T1=T2"
   387     hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp }
   388   moreover
   389   { assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
   390     hence "openin T1 = openin T2" by (simp add: fun_eq_iff)
   391     hence "topology (openin T1) = topology (openin T2)" by simp
   392     hence "T1 = T2" unfolding openin_inverse .
   393   }
   394   ultimately show ?thesis by blast
   395 qed
   396 
   397 text{* Infer the "universe" from union of all sets in the topology. *}
   398 
   399 definition "topspace T =  \<Union>{S. openin T S}"
   400 
   401 subsubsection {* Main properties of open sets *}
   402 
   403 lemma openin_clauses:
   404   fixes U :: "'a topology"
   405   shows "openin U {}"
   406   "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
   407   "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
   408   using openin[of U] unfolding istopology_def mem_Collect_eq
   409   by fast+
   410 
   411 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
   412   unfolding topspace_def by blast
   413 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
   414 
   415 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
   416   using openin_clauses by simp
   417 
   418 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
   419   using openin_clauses by simp
   420 
   421 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
   422   using openin_Union[of "{S,T}" U] by auto
   423 
   424 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
   425 
   426 lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
   427   (is "?lhs \<longleftrightarrow> ?rhs")
   428 proof
   429   assume ?lhs
   430   then show ?rhs by auto
   431 next
   432   assume H: ?rhs
   433   let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
   434   have "openin U ?t" by (simp add: openin_Union)
   435   also have "?t = S" using H by auto
   436   finally show "openin U S" .
   437 qed
   438 
   439 
   440 subsubsection {* Closed sets *}
   441 
   442 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
   443 
   444 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
   445 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
   446 lemma closedin_topspace[intro,simp]:
   447   "closedin U (topspace U)" by (simp add: closedin_def)
   448 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
   449   by (auto simp add: Diff_Un closedin_def)
   450 
   451 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
   452 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
   453   shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
   454 
   455 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
   456   using closedin_Inter[of "{S,T}" U] by auto
   457 
   458 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
   459 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
   460   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
   461   apply (metis openin_subset subset_eq)
   462   done
   463 
   464 lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
   465   by (simp add: openin_closedin_eq)
   466 
   467 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
   468 proof-
   469   have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
   470     by (auto simp add: topspace_def openin_subset)
   471   then show ?thesis using oS cT by (auto simp add: closedin_def)
   472 qed
   473 
   474 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
   475 proof-
   476   have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
   477     by (auto simp add: topspace_def )
   478   then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
   479 qed
   480 
   481 subsubsection {* Subspace topology *}
   482 
   483 definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
   484 
   485 lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
   486   (is "istopology ?L")
   487 proof-
   488   have "?L {}" by blast
   489   {fix A B assume A: "?L A" and B: "?L B"
   490     from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast
   491     have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
   492     then have "?L (A \<inter> B)" by blast}
   493   moreover
   494   {fix K assume K: "K \<subseteq> Collect ?L"
   495     have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
   496       apply (rule set_eqI)
   497       apply (simp add: Ball_def image_iff)
   498       by metis
   499     from K[unfolded th0 subset_image_iff]
   500     obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
   501     have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
   502     moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)
   503     ultimately have "?L (\<Union>K)" by blast}
   504   ultimately show ?thesis
   505     unfolding subset_eq mem_Collect_eq istopology_def by blast
   506 qed
   507 
   508 lemma openin_subtopology:
   509   "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
   510   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
   511   by auto
   512 
   513 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
   514   by (auto simp add: topspace_def openin_subtopology)
   515 
   516 lemma closedin_subtopology:
   517   "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
   518   unfolding closedin_def topspace_subtopology
   519   apply (simp add: openin_subtopology)
   520   apply (rule iffI)
   521   apply clarify
   522   apply (rule_tac x="topspace U - T" in exI)
   523   by auto
   524 
   525 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
   526   unfolding openin_subtopology
   527   apply (rule iffI, clarify)
   528   apply (frule openin_subset[of U])  apply blast
   529   apply (rule exI[where x="topspace U"])
   530   apply auto
   531   done
   532 
   533 lemma subtopology_superset:
   534   assumes UV: "topspace U \<subseteq> V"
   535   shows "subtopology U V = U"
   536 proof-
   537   {fix S
   538     {fix T assume T: "openin U T" "S = T \<inter> V"
   539       from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
   540       have "openin U S" unfolding eq using T by blast}
   541     moreover
   542     {assume S: "openin U S"
   543       hence "\<exists>T. openin U T \<and> S = T \<inter> V"
   544         using openin_subset[OF S] UV by auto}
   545     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
   546   then show ?thesis unfolding topology_eq openin_subtopology by blast
   547 qed
   548 
   549 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
   550   by (simp add: subtopology_superset)
   551 
   552 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
   553   by (simp add: subtopology_superset)
   554 
   555 subsubsection {* The standard Euclidean topology *}
   556 
   557 definition
   558   euclidean :: "'a::topological_space topology" where
   559   "euclidean = topology open"
   560 
   561 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
   562   unfolding euclidean_def
   563   apply (rule cong[where x=S and y=S])
   564   apply (rule topology_inverse[symmetric])
   565   apply (auto simp add: istopology_def)
   566   done
   567 
   568 lemma topspace_euclidean: "topspace euclidean = UNIV"
   569   apply (simp add: topspace_def)
   570   apply (rule set_eqI)
   571   by (auto simp add: open_openin[symmetric])
   572 
   573 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
   574   by (simp add: topspace_euclidean topspace_subtopology)
   575 
   576 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
   577   by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
   578 
   579 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
   580   by (simp add: open_openin openin_subopen[symmetric])
   581 
   582 text {* Basic "localization" results are handy for connectedness. *}
   583 
   584 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
   585   by (auto simp add: openin_subtopology open_openin[symmetric])
   586 
   587 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
   588   by (auto simp add: openin_open)
   589 
   590 lemma open_openin_trans[trans]:
   591  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
   592   by (metis Int_absorb1  openin_open_Int)
   593 
   594 lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
   595   by (auto simp add: openin_open)
   596 
   597 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
   598   by (simp add: closedin_subtopology closed_closedin Int_ac)
   599 
   600 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
   601   by (metis closedin_closed)
   602 
   603 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
   604   apply (subgoal_tac "S \<inter> T = T" )
   605   apply auto
   606   apply (frule closedin_closed_Int[of T S])
   607   by simp
   608 
   609 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
   610   by (auto simp add: closedin_closed)
   611 
   612 lemma openin_euclidean_subtopology_iff:
   613   fixes S U :: "'a::metric_space set"
   614   shows "openin (subtopology euclidean U) S
   615   \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
   616 proof
   617   assume ?lhs thus ?rhs unfolding openin_open open_dist by blast
   618 next
   619   def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
   620   have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
   621     unfolding T_def
   622     apply clarsimp
   623     apply (rule_tac x="d - dist x a" in exI)
   624     apply (clarsimp simp add: less_diff_eq)
   625     apply (erule rev_bexI)
   626     apply (rule_tac x=d in exI, clarify)
   627     apply (erule le_less_trans [OF dist_triangle])
   628     done
   629   assume ?rhs hence 2: "S = U \<inter> T"
   630     unfolding T_def
   631     apply auto
   632     apply (drule (1) bspec, erule rev_bexI)
   633     apply auto
   634     done
   635   from 1 2 show ?lhs
   636     unfolding openin_open open_dist by fast
   637 qed
   638 
   639 text {* These "transitivity" results are handy too *}
   640 
   641 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
   642   \<Longrightarrow> openin (subtopology euclidean U) S"
   643   unfolding open_openin openin_open by blast
   644 
   645 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
   646   by (auto simp add: openin_open intro: openin_trans)
   647 
   648 lemma closedin_trans[trans]:
   649  "closedin (subtopology euclidean T) S \<Longrightarrow>
   650            closedin (subtopology euclidean U) T
   651            ==> closedin (subtopology euclidean U) S"
   652   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
   653 
   654 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
   655   by (auto simp add: closedin_closed intro: closedin_trans)
   656 
   657 
   658 subsection {* Open and closed balls *}
   659 
   660 definition
   661   ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
   662   "ball x e = {y. dist x y < e}"
   663 
   664 definition
   665   cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
   666   "cball x e = {y. dist x y \<le> e}"
   667 
   668 lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
   669   by (simp add: ball_def)
   670 
   671 lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
   672   by (simp add: cball_def)
   673 
   674 lemma mem_ball_0:
   675   fixes x :: "'a::real_normed_vector"
   676   shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
   677   by (simp add: dist_norm)
   678 
   679 lemma mem_cball_0:
   680   fixes x :: "'a::real_normed_vector"
   681   shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
   682   by (simp add: dist_norm)
   683 
   684 lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"
   685   by simp
   686 
   687 lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
   688   by simp
   689 
   690 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
   691 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
   692 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
   693 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
   694   by (simp add: set_eq_iff) arith
   695 
   696 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
   697   by (simp add: set_eq_iff)
   698 
   699 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
   700   "(a::real) - b < 0 \<longleftrightarrow> a < b"
   701   "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
   702 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
   703   "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
   704 
   705 lemma open_ball[intro, simp]: "open (ball x e)"
   706   unfolding open_dist ball_def mem_Collect_eq Ball_def
   707   unfolding dist_commute
   708   apply clarify
   709   apply (rule_tac x="e - dist xa x" in exI)
   710   using dist_triangle_alt[where z=x]
   711   apply (clarsimp simp add: diff_less_iff)
   712   apply atomize
   713   apply (erule_tac x="y" in allE)
   714   apply (erule_tac x="xa" in allE)
   715   by arith
   716 
   717 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
   718   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
   719 
   720 lemma openE[elim?]:
   721   assumes "open S" "x\<in>S" 
   722   obtains e where "e>0" "ball x e \<subseteq> S"
   723   using assms unfolding open_contains_ball by auto
   724 
   725 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
   726   by (metis open_contains_ball subset_eq centre_in_ball)
   727 
   728 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
   729   unfolding mem_ball set_eq_iff
   730   apply (simp add: not_less)
   731   by (metis zero_le_dist order_trans dist_self)
   732 
   733 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
   734 
   735 lemma euclidean_dist_l2:
   736   fixes x y :: "'a :: euclidean_space"
   737   shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
   738   unfolding dist_norm norm_eq_sqrt_inner setL2_def
   739   by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
   740 
   741 definition "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
   742 
   743 lemma rational_boxes:
   744   fixes x :: "'a\<Colon>euclidean_space"
   745   assumes "0 < e"
   746   shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
   747 proof -
   748   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
   749   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive)
   750   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
   751   proof
   752     fix i from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e show "?th i" by auto
   753   qed
   754   from choice[OF this] guess a .. note a = this
   755   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
   756   proof
   757     fix i from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e show "?th i" by auto
   758   qed
   759   from choice[OF this] guess b .. note b = this
   760   let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
   761   show ?thesis
   762   proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
   763     fix y :: 'a assume *: "y \<in> box ?a ?b"
   764     have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<twosuperior>)"
   765       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
   766     also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
   767     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
   768       fix i :: "'a" assume i: "i \<in> Basis"
   769       have "a i < y\<bullet>i \<and> y\<bullet>i < b i" using * i by (auto simp: box_def)
   770       moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'" using a by auto
   771       moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'" using b by auto
   772       ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'" by auto
   773       then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
   774         unfolding e'_def by (auto simp: dist_real_def)
   775       then have "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
   776         by (rule power_strict_mono) auto
   777       then show "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
   778         by (simp add: power_divide)
   779     qed auto
   780     also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat)
   781     finally show "y \<in> ball x e" by (auto simp: ball_def)
   782   qed (insert a b, auto simp: box_def)
   783 qed
   784 
   785 lemma open_UNION_box:
   786   fixes M :: "'a\<Colon>euclidean_space set"
   787   assumes "open M" 
   788   defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
   789   defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
   790   defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^isub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
   791   shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
   792 proof safe
   793   fix x assume "x \<in> M"
   794   obtain e where e: "e > 0" "ball x e \<subseteq> M"
   795     using openE[OF `open M` `x \<in> M`] by auto
   796   moreover then obtain a b where ab: "x \<in> box a b"
   797     "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"
   798     using rational_boxes[OF e(1)] by metis
   799   ultimately show "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"
   800      by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
   801         (auto simp: euclidean_representation I_def a'_def b'_def)
   802 qed (auto simp: I_def)
   803 
   804 subsection{* Connectedness *}
   805 
   806 definition "connected S \<longleftrightarrow>
   807   ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
   808   \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
   809 
   810 lemma connected_local:
   811  "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
   812                  openin (subtopology euclidean S) e1 \<and>
   813                  openin (subtopology euclidean S) e2 \<and>
   814                  S \<subseteq> e1 \<union> e2 \<and>
   815                  e1 \<inter> e2 = {} \<and>
   816                  ~(e1 = {}) \<and>
   817                  ~(e2 = {}))"
   818 unfolding connected_def openin_open by (safe, blast+)
   819 
   820 lemma exists_diff:
   821   fixes P :: "'a set \<Rightarrow> bool"
   822   shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
   823 proof-
   824   {assume "?lhs" hence ?rhs by blast }
   825   moreover
   826   {fix S assume H: "P S"
   827     have "S = - (- S)" by auto
   828     with H have "P (- (- S))" by metis }
   829   ultimately show ?thesis by metis
   830 qed
   831 
   832 lemma connected_clopen: "connected S \<longleftrightarrow>
   833         (\<forall>T. openin (subtopology euclidean S) T \<and>
   834             closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
   835 proof-
   836   have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
   837     unfolding connected_def openin_open closedin_closed
   838     apply (subst exists_diff) by blast
   839   hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
   840     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
   841 
   842   have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
   843     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
   844     unfolding connected_def openin_open closedin_closed by auto
   845   {fix e2
   846     {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
   847         by auto}
   848     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
   849   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
   850   then show ?thesis unfolding th0 th1 by simp
   851 qed
   852 
   853 lemma connected_empty[simp, intro]: "connected {}"
   854   by (simp add: connected_def)
   855 
   856 
   857 subsection{* Limit points *}
   858 
   859 definition (in topological_space)
   860   islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
   861   "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
   862 
   863 lemma islimptI:
   864   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
   865   shows "x islimpt S"
   866   using assms unfolding islimpt_def by auto
   867 
   868 lemma islimptE:
   869   assumes "x islimpt S" and "x \<in> T" and "open T"
   870   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
   871   using assms unfolding islimpt_def by auto
   872 
   873 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
   874   unfolding islimpt_def eventually_at_topological by auto
   875 
   876 lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"
   877   unfolding islimpt_def by fast
   878 
   879 lemma islimpt_approachable:
   880   fixes x :: "'a::metric_space"
   881   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
   882   unfolding islimpt_iff_eventually eventually_at by fast
   883 
   884 lemma islimpt_approachable_le:
   885   fixes x :: "'a::metric_space"
   886   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
   887   unfolding islimpt_approachable
   888   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
   889     THEN arg_cong [where f=Not]]
   890   by (simp add: Bex_def conj_commute conj_left_commute)
   891 
   892 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
   893   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
   894 
   895 text {* A perfect space has no isolated points. *}
   896 
   897 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
   898   unfolding islimpt_UNIV_iff by (rule not_open_singleton)
   899 
   900 lemma perfect_choose_dist:
   901   fixes x :: "'a::{perfect_space, metric_space}"
   902   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
   903 using islimpt_UNIV [of x]
   904 by (simp add: islimpt_approachable)
   905 
   906 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
   907   unfolding closed_def
   908   apply (subst open_subopen)
   909   apply (simp add: islimpt_def subset_eq)
   910   by (metis ComplE ComplI)
   911 
   912 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
   913   unfolding islimpt_def by auto
   914 
   915 lemma finite_set_avoid:
   916   fixes a :: "'a::metric_space"
   917   assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
   918 proof(induct rule: finite_induct[OF fS])
   919   case 1 thus ?case by (auto intro: zero_less_one)
   920 next
   921   case (2 x F)
   922   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
   923   {assume "x = a" hence ?case using d by auto  }
   924   moreover
   925   {assume xa: "x\<noteq>a"
   926     let ?d = "min d (dist a x)"
   927     have dp: "?d > 0" using xa d(1) using dist_nz by auto
   928     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
   929     with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
   930   ultimately show ?case by blast
   931 qed
   932 
   933 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
   934   by (simp add: islimpt_iff_eventually eventually_conj_iff)
   935 
   936 lemma discrete_imp_closed:
   937   fixes S :: "'a::metric_space set"
   938   assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
   939   shows "closed S"
   940 proof-
   941   {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
   942     from e have e2: "e/2 > 0" by arith
   943     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
   944     let ?m = "min (e/2) (dist x y) "
   945     from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
   946     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
   947     have th: "dist z y < e" using z y
   948       by (intro dist_triangle_lt [where z=x], simp)
   949     from d[rule_format, OF y(1) z(1) th] y z
   950     have False by (auto simp add: dist_commute)}
   951   then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
   952 qed
   953 
   954 
   955 subsection {* Interior of a Set *}
   956 
   957 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
   958 
   959 lemma interiorI [intro?]:
   960   assumes "open T" and "x \<in> T" and "T \<subseteq> S"
   961   shows "x \<in> interior S"
   962   using assms unfolding interior_def by fast
   963 
   964 lemma interiorE [elim?]:
   965   assumes "x \<in> interior S"
   966   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
   967   using assms unfolding interior_def by fast
   968 
   969 lemma open_interior [simp, intro]: "open (interior S)"
   970   by (simp add: interior_def open_Union)
   971 
   972 lemma interior_subset: "interior S \<subseteq> S"
   973   by (auto simp add: interior_def)
   974 
   975 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
   976   by (auto simp add: interior_def)
   977 
   978 lemma interior_open: "open S \<Longrightarrow> interior S = S"
   979   by (intro equalityI interior_subset interior_maximal subset_refl)
   980 
   981 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
   982   by (metis open_interior interior_open)
   983 
   984 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
   985   by (metis interior_maximal interior_subset subset_trans)
   986 
   987 lemma interior_empty [simp]: "interior {} = {}"
   988   using open_empty by (rule interior_open)
   989 
   990 lemma interior_UNIV [simp]: "interior UNIV = UNIV"
   991   using open_UNIV by (rule interior_open)
   992 
   993 lemma interior_interior [simp]: "interior (interior S) = interior S"
   994   using open_interior by (rule interior_open)
   995 
   996 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
   997   by (auto simp add: interior_def)
   998 
   999 lemma interior_unique:
  1000   assumes "T \<subseteq> S" and "open T"
  1001   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
  1002   shows "interior S = T"
  1003   by (intro equalityI assms interior_subset open_interior interior_maximal)
  1004 
  1005 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
  1006   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
  1007     Int_lower2 interior_maximal interior_subset open_Int open_interior)
  1008 
  1009 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
  1010   using open_contains_ball_eq [where S="interior S"]
  1011   by (simp add: open_subset_interior)
  1012 
  1013 lemma interior_limit_point [intro]:
  1014   fixes x :: "'a::perfect_space"
  1015   assumes x: "x \<in> interior S" shows "x islimpt S"
  1016   using x islimpt_UNIV [of x]
  1017   unfolding interior_def islimpt_def
  1018   apply (clarsimp, rename_tac T T')
  1019   apply (drule_tac x="T \<inter> T'" in spec)
  1020   apply (auto simp add: open_Int)
  1021   done
  1022 
  1023 lemma interior_closed_Un_empty_interior:
  1024   assumes cS: "closed S" and iT: "interior T = {}"
  1025   shows "interior (S \<union> T) = interior S"
  1026 proof
  1027   show "interior S \<subseteq> interior (S \<union> T)"
  1028     by (rule interior_mono, rule Un_upper1)
  1029 next
  1030   show "interior (S \<union> T) \<subseteq> interior S"
  1031   proof
  1032     fix x assume "x \<in> interior (S \<union> T)"
  1033     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
  1034     show "x \<in> interior S"
  1035     proof (rule ccontr)
  1036       assume "x \<notin> interior S"
  1037       with `x \<in> R` `open R` obtain y where "y \<in> R - S"
  1038         unfolding interior_def by fast
  1039       from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
  1040       from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
  1041       from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
  1042       show "False" unfolding interior_def by fast
  1043     qed
  1044   qed
  1045 qed
  1046 
  1047 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
  1048 proof (rule interior_unique)
  1049   show "interior A \<times> interior B \<subseteq> A \<times> B"
  1050     by (intro Sigma_mono interior_subset)
  1051   show "open (interior A \<times> interior B)"
  1052     by (intro open_Times open_interior)
  1053   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"
  1054   proof (safe)
  1055     fix x y assume "(x, y) \<in> T"
  1056     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
  1057       using `open T` unfolding open_prod_def by fast
  1058     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
  1059       using `T \<subseteq> A \<times> B` by auto
  1060     thus "x \<in> interior A" and "y \<in> interior B"
  1061       by (auto intro: interiorI)
  1062   qed
  1063 qed
  1064 
  1065 
  1066 subsection {* Closure of a Set *}
  1067 
  1068 definition "closure S = S \<union> {x | x. x islimpt S}"
  1069 
  1070 lemma interior_closure: "interior S = - (closure (- S))"
  1071   unfolding interior_def closure_def islimpt_def by auto
  1072 
  1073 lemma closure_interior: "closure S = - interior (- S)"
  1074   unfolding interior_closure by simp
  1075 
  1076 lemma closed_closure[simp, intro]: "closed (closure S)"
  1077   unfolding closure_interior by (simp add: closed_Compl)
  1078 
  1079 lemma closure_subset: "S \<subseteq> closure S"
  1080   unfolding closure_def by simp
  1081 
  1082 lemma closure_hull: "closure S = closed hull S"
  1083   unfolding hull_def closure_interior interior_def by auto
  1084 
  1085 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
  1086   unfolding closure_hull using closed_Inter by (rule hull_eq)
  1087 
  1088 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
  1089   unfolding closure_eq .
  1090 
  1091 lemma closure_closure [simp]: "closure (closure S) = closure S"
  1092   unfolding closure_hull by (rule hull_hull)
  1093 
  1094 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
  1095   unfolding closure_hull by (rule hull_mono)
  1096 
  1097 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
  1098   unfolding closure_hull by (rule hull_minimal)
  1099 
  1100 lemma closure_unique:
  1101   assumes "S \<subseteq> T" and "closed T"
  1102   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
  1103   shows "closure S = T"
  1104   using assms unfolding closure_hull by (rule hull_unique)
  1105 
  1106 lemma closure_empty [simp]: "closure {} = {}"
  1107   using closed_empty by (rule closure_closed)
  1108 
  1109 lemma closure_UNIV [simp]: "closure UNIV = UNIV"
  1110   using closed_UNIV by (rule closure_closed)
  1111 
  1112 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
  1113   unfolding closure_interior by simp
  1114 
  1115 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
  1116   using closure_empty closure_subset[of S]
  1117   by blast
  1118 
  1119 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
  1120   using closure_eq[of S] closure_subset[of S]
  1121   by simp
  1122 
  1123 lemma open_inter_closure_eq_empty:
  1124   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
  1125   using open_subset_interior[of S "- T"]
  1126   using interior_subset[of "- T"]
  1127   unfolding closure_interior
  1128   by auto
  1129 
  1130 lemma open_inter_closure_subset:
  1131   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
  1132 proof
  1133   fix x
  1134   assume as: "open S" "x \<in> S \<inter> closure T"
  1135   { assume *:"x islimpt T"
  1136     have "x islimpt (S \<inter> T)"
  1137     proof (rule islimptI)
  1138       fix A
  1139       assume "x \<in> A" "open A"
  1140       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
  1141         by (simp_all add: open_Int)
  1142       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
  1143         by (rule islimptE)
  1144       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
  1145         by simp_all
  1146       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
  1147     qed
  1148   }
  1149   then show "x \<in> closure (S \<inter> T)" using as
  1150     unfolding closure_def
  1151     by blast
  1152 qed
  1153 
  1154 lemma closure_complement: "closure (- S) = - interior S"
  1155   unfolding closure_interior by simp
  1156 
  1157 lemma interior_complement: "interior (- S) = - closure S"
  1158   unfolding closure_interior by simp
  1159 
  1160 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
  1161 proof (rule closure_unique)
  1162   show "A \<times> B \<subseteq> closure A \<times> closure B"
  1163     by (intro Sigma_mono closure_subset)
  1164   show "closed (closure A \<times> closure B)"
  1165     by (intro closed_Times closed_closure)
  1166   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"
  1167     apply (simp add: closed_def open_prod_def, clarify)
  1168     apply (rule ccontr)
  1169     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
  1170     apply (simp add: closure_interior interior_def)
  1171     apply (drule_tac x=C in spec)
  1172     apply (drule_tac x=D in spec)
  1173     apply auto
  1174     done
  1175 qed
  1176 
  1177 
  1178 subsection {* Frontier (aka boundary) *}
  1179 
  1180 definition "frontier S = closure S - interior S"
  1181 
  1182 lemma frontier_closed: "closed(frontier S)"
  1183   by (simp add: frontier_def closed_Diff)
  1184 
  1185 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
  1186   by (auto simp add: frontier_def interior_closure)
  1187 
  1188 lemma frontier_straddle:
  1189   fixes a :: "'a::metric_space"
  1190   shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
  1191   unfolding frontier_def closure_interior
  1192   by (auto simp add: mem_interior subset_eq ball_def)
  1193 
  1194 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
  1195   by (metis frontier_def closure_closed Diff_subset)
  1196 
  1197 lemma frontier_empty[simp]: "frontier {} = {}"
  1198   by (simp add: frontier_def)
  1199 
  1200 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
  1201 proof-
  1202   { assume "frontier S \<subseteq> S"
  1203     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
  1204     hence "closed S" using closure_subset_eq by auto
  1205   }
  1206   thus ?thesis using frontier_subset_closed[of S] ..
  1207 qed
  1208 
  1209 lemma frontier_complement: "frontier(- S) = frontier S"
  1210   by (auto simp add: frontier_def closure_complement interior_complement)
  1211 
  1212 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
  1213   using frontier_complement frontier_subset_eq[of "- S"]
  1214   unfolding open_closed by auto
  1215 
  1216 subsection {* Filters and the ``eventually true'' quantifier *}
  1217 
  1218 definition
  1219   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
  1220     (infixr "indirection" 70) where
  1221   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
  1222 
  1223 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
  1224 
  1225 lemma trivial_limit_within:
  1226   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
  1227 proof
  1228   assume "trivial_limit (at a within S)"
  1229   thus "\<not> a islimpt S"
  1230     unfolding trivial_limit_def
  1231     unfolding eventually_within eventually_at_topological
  1232     unfolding islimpt_def
  1233     apply (clarsimp simp add: set_eq_iff)
  1234     apply (rename_tac T, rule_tac x=T in exI)
  1235     apply (clarsimp, drule_tac x=y in bspec, simp_all)
  1236     done
  1237 next
  1238   assume "\<not> a islimpt S"
  1239   thus "trivial_limit (at a within S)"
  1240     unfolding trivial_limit_def
  1241     unfolding eventually_within eventually_at_topological
  1242     unfolding islimpt_def
  1243     apply clarsimp
  1244     apply (rule_tac x=T in exI)
  1245     apply auto
  1246     done
  1247 qed
  1248 
  1249 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
  1250   using trivial_limit_within [of a UNIV] by simp
  1251 
  1252 lemma trivial_limit_at:
  1253   fixes a :: "'a::perfect_space"
  1254   shows "\<not> trivial_limit (at a)"
  1255   by (rule at_neq_bot)
  1256 
  1257 lemma trivial_limit_at_infinity:
  1258   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
  1259   unfolding trivial_limit_def eventually_at_infinity
  1260   apply clarsimp
  1261   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
  1262    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
  1263   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
  1264   apply (drule_tac x=UNIV in spec, simp)
  1265   done
  1266 
  1267 text {* Some property holds "sufficiently close" to the limit point. *}
  1268 
  1269 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
  1270   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
  1271 unfolding eventually_at dist_nz by auto
  1272 
  1273 lemma eventually_within: (* FIXME: this replaces Limits.eventually_within *)
  1274   "eventually P (at a within S) \<longleftrightarrow>
  1275         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
  1276   by (rule eventually_within_less)
  1277 
  1278 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
  1279   unfolding trivial_limit_def
  1280   by (auto elim: eventually_rev_mp)
  1281 
  1282 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
  1283   by simp
  1284 
  1285 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
  1286   by (simp add: filter_eq_iff)
  1287 
  1288 text{* Combining theorems for "eventually" *}
  1289 
  1290 lemma eventually_rev_mono:
  1291   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
  1292 using eventually_mono [of P Q] by fast
  1293 
  1294 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
  1295   by (simp add: eventually_False)
  1296 
  1297 
  1298 subsection {* Limits *}
  1299 
  1300 text{* Notation Lim to avoid collition with lim defined in analysis *}
  1301 
  1302 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
  1303   where "Lim A f = (THE l. (f ---> l) A)"
  1304 
  1305 lemma Lim:
  1306  "(f ---> l) net \<longleftrightarrow>
  1307         trivial_limit net \<or>
  1308         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
  1309   unfolding tendsto_iff trivial_limit_eq by auto
  1310 
  1311 text{* Show that they yield usual definitions in the various cases. *}
  1312 
  1313 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
  1314            (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"
  1315   by (auto simp add: tendsto_iff eventually_within_le)
  1316 
  1317 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
  1318         (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
  1319   by (auto simp add: tendsto_iff eventually_within)
  1320 
  1321 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
  1322         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
  1323   by (auto simp add: tendsto_iff eventually_at)
  1324 
  1325 lemma Lim_at_infinity:
  1326   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
  1327   by (auto simp add: tendsto_iff eventually_at_infinity)
  1328 
  1329 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
  1330   by (rule topological_tendstoI, auto elim: eventually_rev_mono)
  1331 
  1332 text{* The expected monotonicity property. *}
  1333 
  1334 lemma Lim_within_empty: "(f ---> l) (net within {})"
  1335   unfolding tendsto_def Limits.eventually_within by simp
  1336 
  1337 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
  1338   unfolding tendsto_def Limits.eventually_within
  1339   by (auto elim!: eventually_elim1)
  1340 
  1341 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
  1342   shows "(f ---> l) (net within (S \<union> T))"
  1343   using assms unfolding tendsto_def Limits.eventually_within
  1344   apply clarify
  1345   apply (drule spec, drule (1) mp, drule (1) mp)
  1346   apply (drule spec, drule (1) mp, drule (1) mp)
  1347   apply (auto elim: eventually_elim2)
  1348   done
  1349 
  1350 lemma Lim_Un_univ:
  1351  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
  1352         ==> (f ---> l) net"
  1353   by (metis Lim_Un within_UNIV)
  1354 
  1355 text{* Interrelations between restricted and unrestricted limits. *}
  1356 
  1357 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
  1358   (* FIXME: rename *)
  1359   unfolding tendsto_def Limits.eventually_within
  1360   apply (clarify, drule spec, drule (1) mp, drule (1) mp)
  1361   by (auto elim!: eventually_elim1)
  1362 
  1363 lemma eventually_within_interior:
  1364   assumes "x \<in> interior S"
  1365   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
  1366 proof-
  1367   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
  1368   { assume "?lhs"
  1369     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
  1370       unfolding Limits.eventually_within Limits.eventually_at_topological
  1371       by auto
  1372     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
  1373       by auto
  1374     then have "?rhs"
  1375       unfolding Limits.eventually_at_topological by auto
  1376   } moreover
  1377   { assume "?rhs" hence "?lhs"
  1378       unfolding Limits.eventually_within
  1379       by (auto elim: eventually_elim1)
  1380   } ultimately
  1381   show "?thesis" ..
  1382 qed
  1383 
  1384 lemma at_within_interior:
  1385   "x \<in> interior S \<Longrightarrow> at x within S = at x"
  1386   by (simp add: filter_eq_iff eventually_within_interior)
  1387 
  1388 lemma at_within_open:
  1389   "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
  1390   by (simp only: at_within_interior interior_open)
  1391 
  1392 lemma Lim_within_open:
  1393   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
  1394   assumes"a \<in> S" "open S"
  1395   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
  1396   using assms by (simp only: at_within_open)
  1397 
  1398 lemma Lim_within_LIMSEQ:
  1399   fixes a :: "'a::metric_space"
  1400   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
  1401   shows "(X ---> L) (at a within T)"
  1402   using assms unfolding tendsto_def [where l=L]
  1403   by (simp add: sequentially_imp_eventually_within)
  1404 
  1405 lemma Lim_right_bound:
  1406   fixes f :: "real \<Rightarrow> real"
  1407   assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
  1408   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
  1409   shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
  1410 proof cases
  1411   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
  1412 next
  1413   assume [simp]: "{x<..} \<inter> I \<noteq> {}"
  1414   show ?thesis
  1415   proof (rule Lim_within_LIMSEQ, safe)
  1416     fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
  1417     
  1418     show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
  1419     proof (rule LIMSEQ_I, rule ccontr)
  1420       fix r :: real assume "0 < r"
  1421       with Inf_close[of "f ` ({x<..} \<inter> I)" r]
  1422       obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
  1423       from `x < y` have "0 < y - x" by auto
  1424       from S(2)[THEN LIMSEQ_D, OF this]
  1425       obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
  1426       
  1427       assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
  1428       moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
  1429         using S bnd by (intro Inf_lower[where z=K]) auto
  1430       ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
  1431         by (auto simp: not_less field_simps)
  1432       with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
  1433       show False by auto
  1434     qed
  1435   qed
  1436 qed
  1437 
  1438 text{* Another limit point characterization. *}
  1439 
  1440 lemma islimpt_sequential:
  1441   fixes x :: "'a::first_countable_topology"
  1442   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"
  1443     (is "?lhs = ?rhs")
  1444 proof
  1445   assume ?lhs
  1446   from countable_basis_at_decseq[of x] guess A . note A = this
  1447   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
  1448   { fix n
  1449     from `?lhs` have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
  1450       unfolding islimpt_def using A(1,2)[of n] by auto
  1451     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
  1452       unfolding f_def by (rule someI_ex)
  1453     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto }
  1454   then have "\<forall>n. f n \<in> S - {x}" by auto
  1455   moreover have "(\<lambda>n. f n) ----> x"
  1456   proof (rule topological_tendstoI)
  1457     fix S assume "open S" "x \<in> S"
  1458     from A(3)[OF this] `\<And>n. f n \<in> A n`
  1459     show "eventually (\<lambda>x. f x \<in> S) sequentially" by (auto elim!: eventually_elim1)
  1460   qed
  1461   ultimately show ?rhs by fast
  1462 next
  1463   assume ?rhs
  1464   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x" by auto
  1465   show ?lhs
  1466     unfolding islimpt_def
  1467   proof safe
  1468     fix T assume "open T" "x \<in> T"
  1469     from lim[THEN topological_tendstoD, OF this] f
  1470     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
  1471       unfolding eventually_sequentially by auto
  1472   qed
  1473 qed
  1474 
  1475 lemma Lim_inv: (* TODO: delete *)
  1476   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
  1477   assumes "(f ---> l) A" and "l \<noteq> 0"
  1478   shows "((inverse o f) ---> inverse l) A"
  1479   unfolding o_def using assms by (rule tendsto_inverse)
  1480 
  1481 lemma Lim_null:
  1482   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1483   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
  1484   by (simp add: Lim dist_norm)
  1485 
  1486 lemma Lim_null_comparison:
  1487   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1488   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
  1489   shows "(f ---> 0) net"
  1490 proof (rule metric_tendsto_imp_tendsto)
  1491   show "(g ---> 0) net" by fact
  1492   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
  1493     using assms(1) by (rule eventually_elim1, simp add: dist_norm)
  1494 qed
  1495 
  1496 lemma Lim_transform_bound:
  1497   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1498   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
  1499   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
  1500   shows "(f ---> 0) net"
  1501   using assms(1) tendsto_norm_zero [OF assms(2)]
  1502   by (rule Lim_null_comparison)
  1503 
  1504 text{* Deducing things about the limit from the elements. *}
  1505 
  1506 lemma Lim_in_closed_set:
  1507   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
  1508   shows "l \<in> S"
  1509 proof (rule ccontr)
  1510   assume "l \<notin> S"
  1511   with `closed S` have "open (- S)" "l \<in> - S"
  1512     by (simp_all add: open_Compl)
  1513   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
  1514     by (rule topological_tendstoD)
  1515   with assms(2) have "eventually (\<lambda>x. False) net"
  1516     by (rule eventually_elim2) simp
  1517   with assms(3) show "False"
  1518     by (simp add: eventually_False)
  1519 qed
  1520 
  1521 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
  1522 
  1523 lemma Lim_dist_ubound:
  1524   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
  1525   shows "dist a l <= e"
  1526 proof-
  1527   have "dist a l \<in> {..e}"
  1528   proof (rule Lim_in_closed_set)
  1529     show "closed {..e}" by simp
  1530     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
  1531     show "\<not> trivial_limit net" by fact
  1532     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
  1533   qed
  1534   thus ?thesis by simp
  1535 qed
  1536 
  1537 lemma Lim_norm_ubound:
  1538   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1539   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
  1540   shows "norm(l) <= e"
  1541 proof-
  1542   have "norm l \<in> {..e}"
  1543   proof (rule Lim_in_closed_set)
  1544     show "closed {..e}" by simp
  1545     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
  1546     show "\<not> trivial_limit net" by fact
  1547     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
  1548   qed
  1549   thus ?thesis by simp
  1550 qed
  1551 
  1552 lemma Lim_norm_lbound:
  1553   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
  1554   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
  1555   shows "e \<le> norm l"
  1556 proof-
  1557   have "norm l \<in> {e..}"
  1558   proof (rule Lim_in_closed_set)
  1559     show "closed {e..}" by simp
  1560     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
  1561     show "\<not> trivial_limit net" by fact
  1562     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
  1563   qed
  1564   thus ?thesis by simp
  1565 qed
  1566 
  1567 text{* Uniqueness of the limit, when nontrivial. *}
  1568 
  1569 lemma tendsto_Lim:
  1570   fixes f :: "'a \<Rightarrow> 'b::t2_space"
  1571   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
  1572   unfolding Lim_def using tendsto_unique[of net f] by auto
  1573 
  1574 text{* Limit under bilinear function *}
  1575 
  1576 lemma Lim_bilinear:
  1577   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
  1578   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
  1579 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
  1580 by (rule bounded_bilinear.tendsto)
  1581 
  1582 text{* These are special for limits out of the same vector space. *}
  1583 
  1584 lemma Lim_within_id: "(id ---> a) (at a within s)"
  1585   unfolding id_def by (rule tendsto_ident_at_within)
  1586 
  1587 lemma Lim_at_id: "(id ---> a) (at a)"
  1588   unfolding id_def by (rule tendsto_ident_at)
  1589 
  1590 lemma Lim_at_zero:
  1591   fixes a :: "'a::real_normed_vector"
  1592   fixes l :: "'b::topological_space"
  1593   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
  1594   using LIM_offset_zero LIM_offset_zero_cancel ..
  1595 
  1596 text{* It's also sometimes useful to extract the limit point from the filter. *}
  1597 
  1598 definition
  1599   netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
  1600   "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
  1601 
  1602 lemma netlimit_within:
  1603   assumes "\<not> trivial_limit (at a within S)"
  1604   shows "netlimit (at a within S) = a"
  1605 unfolding netlimit_def
  1606 apply (rule some_equality)
  1607 apply (rule Lim_at_within)
  1608 apply (rule tendsto_ident_at)
  1609 apply (erule tendsto_unique [OF assms])
  1610 apply (rule Lim_at_within)
  1611 apply (rule tendsto_ident_at)
  1612 done
  1613 
  1614 lemma netlimit_at:
  1615   fixes a :: "'a::{perfect_space,t2_space}"
  1616   shows "netlimit (at a) = a"
  1617   using netlimit_within [of a UNIV] by simp
  1618 
  1619 lemma lim_within_interior:
  1620   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
  1621   by (simp add: at_within_interior)
  1622 
  1623 lemma netlimit_within_interior:
  1624   fixes x :: "'a::{t2_space,perfect_space}"
  1625   assumes "x \<in> interior S"
  1626   shows "netlimit (at x within S) = x"
  1627 using assms by (simp add: at_within_interior netlimit_at)
  1628 
  1629 text{* Transformation of limit. *}
  1630 
  1631 lemma Lim_transform:
  1632   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
  1633   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
  1634   shows "(g ---> l) net"
  1635   using tendsto_diff [OF assms(2) assms(1)] by simp
  1636 
  1637 lemma Lim_transform_eventually:
  1638   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
  1639   apply (rule topological_tendstoI)
  1640   apply (drule (2) topological_tendstoD)
  1641   apply (erule (1) eventually_elim2, simp)
  1642   done
  1643 
  1644 lemma Lim_transform_within:
  1645   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  1646   and "(f ---> l) (at x within S)"
  1647   shows "(g ---> l) (at x within S)"
  1648 proof (rule Lim_transform_eventually)
  1649   show "eventually (\<lambda>x. f x = g x) (at x within S)"
  1650     unfolding eventually_within
  1651     using assms(1,2) by auto
  1652   show "(f ---> l) (at x within S)" by fact
  1653 qed
  1654 
  1655 lemma Lim_transform_at:
  1656   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  1657   and "(f ---> l) (at x)"
  1658   shows "(g ---> l) (at x)"
  1659 proof (rule Lim_transform_eventually)
  1660   show "eventually (\<lambda>x. f x = g x) (at x)"
  1661     unfolding eventually_at
  1662     using assms(1,2) by auto
  1663   show "(f ---> l) (at x)" by fact
  1664 qed
  1665 
  1666 text{* Common case assuming being away from some crucial point like 0. *}
  1667 
  1668 lemma Lim_transform_away_within:
  1669   fixes a b :: "'a::t1_space"
  1670   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1671   and "(f ---> l) (at a within S)"
  1672   shows "(g ---> l) (at a within S)"
  1673 proof (rule Lim_transform_eventually)
  1674   show "(f ---> l) (at a within S)" by fact
  1675   show "eventually (\<lambda>x. f x = g x) (at a within S)"
  1676     unfolding Limits.eventually_within eventually_at_topological
  1677     by (rule exI [where x="- {b}"], simp add: open_Compl assms)
  1678 qed
  1679 
  1680 lemma Lim_transform_away_at:
  1681   fixes a b :: "'a::t1_space"
  1682   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1683   and fl: "(f ---> l) (at a)"
  1684   shows "(g ---> l) (at a)"
  1685   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
  1686   by simp
  1687 
  1688 text{* Alternatively, within an open set. *}
  1689 
  1690 lemma Lim_transform_within_open:
  1691   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
  1692   and "(f ---> l) (at a)"
  1693   shows "(g ---> l) (at a)"
  1694 proof (rule Lim_transform_eventually)
  1695   show "eventually (\<lambda>x. f x = g x) (at a)"
  1696     unfolding eventually_at_topological
  1697     using assms(1,2,3) by auto
  1698   show "(f ---> l) (at a)" by fact
  1699 qed
  1700 
  1701 text{* A congruence rule allowing us to transform limits assuming not at point. *}
  1702 
  1703 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
  1704 
  1705 lemma Lim_cong_within(*[cong add]*):
  1706   assumes "a = b" "x = y" "S = T"
  1707   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
  1708   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
  1709   unfolding tendsto_def Limits.eventually_within eventually_at_topological
  1710   using assms by simp
  1711 
  1712 lemma Lim_cong_at(*[cong add]*):
  1713   assumes "a = b" "x = y"
  1714   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
  1715   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
  1716   unfolding tendsto_def eventually_at_topological
  1717   using assms by simp
  1718 
  1719 text{* Useful lemmas on closure and set of possible sequential limits.*}
  1720 
  1721 lemma closure_sequential:
  1722   fixes l :: "'a::first_countable_topology"
  1723   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
  1724 proof
  1725   assume "?lhs" moreover
  1726   { assume "l \<in> S"
  1727     hence "?rhs" using tendsto_const[of l sequentially] by auto
  1728   } moreover
  1729   { assume "l islimpt S"
  1730     hence "?rhs" unfolding islimpt_sequential by auto
  1731   } ultimately
  1732   show "?rhs" unfolding closure_def by auto
  1733 next
  1734   assume "?rhs"
  1735   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
  1736 qed
  1737 
  1738 lemma closed_sequential_limits:
  1739   fixes S :: "'a::first_countable_topology set"
  1740   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
  1741   unfolding closed_limpt
  1742   using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]
  1743   by metis
  1744 
  1745 lemma closure_approachable:
  1746   fixes S :: "'a::metric_space set"
  1747   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
  1748   apply (auto simp add: closure_def islimpt_approachable)
  1749   by (metis dist_self)
  1750 
  1751 lemma closed_approachable:
  1752   fixes S :: "'a::metric_space set"
  1753   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
  1754   by (metis closure_closed closure_approachable)
  1755 
  1756 subsection {* Infimum Distance *}
  1757 
  1758 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"
  1759 
  1760 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"
  1761   by (simp add: infdist_def)
  1762 
  1763 lemma infdist_nonneg:
  1764   shows "0 \<le> infdist x A"
  1765   using assms by (auto simp add: infdist_def)
  1766 
  1767 lemma infdist_le:
  1768   assumes "a \<in> A"
  1769   assumes "d = dist x a"
  1770   shows "infdist x A \<le> d"
  1771   using assms by (auto intro!: SupInf.Inf_lower[where z=0] simp add: infdist_def)
  1772 
  1773 lemma infdist_zero[simp]:
  1774   assumes "a \<in> A" shows "infdist a A = 0"
  1775 proof -
  1776   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto
  1777   with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto
  1778 qed
  1779 
  1780 lemma infdist_triangle:
  1781   shows "infdist x A \<le> infdist y A + dist x y"
  1782 proof cases
  1783   assume "A = {}" thus ?thesis by (simp add: infdist_def)
  1784 next
  1785   assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto
  1786   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"
  1787   proof
  1788     from `A \<noteq> {}` show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp
  1789     fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"
  1790     then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto
  1791     show "infdist x A \<le> d"
  1792       unfolding infdist_notempty[OF `A \<noteq> {}`]
  1793     proof (rule Inf_lower2)
  1794       show "dist x a \<in> {dist x a |a. a \<in> A}" using `a \<in> A` by auto
  1795       show "dist x a \<le> d" unfolding d by (rule dist_triangle)
  1796       fix d assume "d \<in> {dist x a |a. a \<in> A}"
  1797       then obtain a where "a \<in> A" "d = dist x a" by auto
  1798       thus "infdist x A \<le> d" by (rule infdist_le)
  1799     qed
  1800   qed
  1801   also have "\<dots> = dist x y + infdist y A"
  1802   proof (rule Inf_eq, safe)
  1803     fix a assume "a \<in> A"
  1804     thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)
  1805   next
  1806     fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
  1807     hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF `A \<noteq> {}`] using `a \<in> A`
  1808       by (intro Inf_greatest) (auto simp: field_simps)
  1809     thus "i \<le> dist x y + infdist y A" by simp
  1810   qed
  1811   finally show ?thesis by simp
  1812 qed
  1813 
  1814 lemma
  1815   in_closure_iff_infdist_zero:
  1816   assumes "A \<noteq> {}"
  1817   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
  1818 proof
  1819   assume "x \<in> closure A"
  1820   show "infdist x A = 0"
  1821   proof (rule ccontr)
  1822     assume "infdist x A \<noteq> 0"
  1823     with infdist_nonneg[of x A] have "infdist x A > 0" by auto
  1824     hence "ball x (infdist x A) \<inter> closure A = {}" apply auto
  1825       by (metis `0 < infdist x A` `x \<in> closure A` closure_approachable dist_commute
  1826         eucl_less_not_refl euclidean_trans(2) infdist_le)
  1827     hence "x \<notin> closure A" by (metis `0 < infdist x A` centre_in_ball disjoint_iff_not_equal)
  1828     thus False using `x \<in> closure A` by simp
  1829   qed
  1830 next
  1831   assume x: "infdist x A = 0"
  1832   then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)
  1833   show "x \<in> closure A" unfolding closure_approachable
  1834   proof (safe, rule ccontr)
  1835     fix e::real assume "0 < e"
  1836     assume "\<not> (\<exists>y\<in>A. dist y x < e)"
  1837     hence "infdist x A \<ge> e" using `a \<in> A`
  1838       unfolding infdist_def
  1839       by (force simp: dist_commute)
  1840     with x `0 < e` show False by auto
  1841   qed
  1842 qed
  1843 
  1844 lemma
  1845   in_closed_iff_infdist_zero:
  1846   assumes "closed A" "A \<noteq> {}"
  1847   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"
  1848 proof -
  1849   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
  1850     by (rule in_closure_iff_infdist_zero) fact
  1851   with assms show ?thesis by simp
  1852 qed
  1853 
  1854 lemma tendsto_infdist [tendsto_intros]:
  1855   assumes f: "(f ---> l) F"
  1856   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"
  1857 proof (rule tendstoI)
  1858   fix e ::real assume "0 < e"
  1859   from tendstoD[OF f this]
  1860   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"
  1861   proof (eventually_elim)
  1862     fix x
  1863     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
  1864     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"
  1865       by (simp add: dist_commute dist_real_def)
  1866     also assume "dist (f x) l < e"
  1867     finally show "dist (infdist (f x) A) (infdist l A) < e" .
  1868   qed
  1869 qed
  1870 
  1871 text{* Some other lemmas about sequences. *}
  1872 
  1873 lemma sequentially_offset:
  1874   assumes "eventually (\<lambda>i. P i) sequentially"
  1875   shows "eventually (\<lambda>i. P (i + k)) sequentially"
  1876   using assms unfolding eventually_sequentially by (metis trans_le_add1)
  1877 
  1878 lemma seq_offset:
  1879   assumes "(f ---> l) sequentially"
  1880   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
  1881   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)
  1882 
  1883 lemma seq_offset_neg:
  1884   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
  1885   apply (rule topological_tendstoI)
  1886   apply (drule (2) topological_tendstoD)
  1887   apply (simp only: eventually_sequentially)
  1888   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
  1889   apply metis
  1890   by arith
  1891 
  1892 lemma seq_offset_rev:
  1893   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
  1894   by (rule LIMSEQ_offset) (* FIXME: redundant *)
  1895 
  1896 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
  1897   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)
  1898 
  1899 subsection {* More properties of closed balls *}
  1900 
  1901 lemma closed_cball: "closed (cball x e)"
  1902 unfolding cball_def closed_def
  1903 unfolding Collect_neg_eq [symmetric] not_le
  1904 apply (clarsimp simp add: open_dist, rename_tac y)
  1905 apply (rule_tac x="dist x y - e" in exI, clarsimp)
  1906 apply (rename_tac x')
  1907 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
  1908 apply simp
  1909 done
  1910 
  1911 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
  1912 proof-
  1913   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
  1914     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
  1915   } moreover
  1916   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
  1917     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
  1918   } ultimately
  1919   show ?thesis unfolding open_contains_ball by auto
  1920 qed
  1921 
  1922 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
  1923   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
  1924 
  1925 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
  1926   apply (simp add: interior_def, safe)
  1927   apply (force simp add: open_contains_cball)
  1928   apply (rule_tac x="ball x e" in exI)
  1929   apply (simp add: subset_trans [OF ball_subset_cball])
  1930   done
  1931 
  1932 lemma islimpt_ball:
  1933   fixes x y :: "'a::{real_normed_vector,perfect_space}"
  1934   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
  1935 proof
  1936   assume "?lhs"
  1937   { assume "e \<le> 0"
  1938     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
  1939     have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
  1940   }
  1941   hence "e > 0" by (metis not_less)
  1942   moreover
  1943   have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] `?lhs` unfolding closed_limpt by auto
  1944   ultimately show "?rhs" by auto
  1945 next
  1946   assume "?rhs" hence "e>0"  by auto
  1947   { fix d::real assume "d>0"
  1948     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1949     proof(cases "d \<le> dist x y")
  1950       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1951       proof(cases "x=y")
  1952         case True hence False using `d \<le> dist x y` `d>0` by auto
  1953         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
  1954       next
  1955         case False
  1956 
  1957         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
  1958               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  1959           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
  1960         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
  1961           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
  1962           unfolding scaleR_minus_left scaleR_one
  1963           by (auto simp add: norm_minus_commute)
  1964         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
  1965           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
  1966           unfolding distrib_right using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
  1967         also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
  1968         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
  1969 
  1970         moreover
  1971 
  1972         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
  1973           using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
  1974         moreover
  1975         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
  1976           using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
  1977           unfolding dist_norm by auto
  1978         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac  x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
  1979       qed
  1980     next
  1981       case False hence "d > dist x y" by auto
  1982       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1983       proof(cases "x=y")
  1984         case True
  1985         obtain z where **: "z \<noteq> y" "dist z y < min e d"
  1986           using perfect_choose_dist[of "min e d" y]
  1987           using `d > 0` `e>0` by auto
  1988         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1989           unfolding `x = y`
  1990           using `z \<noteq> y` **
  1991           by (rule_tac x=z in bexI, auto simp add: dist_commute)
  1992       next
  1993         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
  1994           using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
  1995       qed
  1996     qed  }
  1997   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
  1998 qed
  1999 
  2000 lemma closure_ball_lemma:
  2001   fixes x y :: "'a::real_normed_vector"
  2002   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
  2003 proof (rule islimptI)
  2004   fix T assume "y \<in> T" "open T"
  2005   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
  2006     unfolding open_dist by fast
  2007   (* choose point between x and y, within distance r of y. *)
  2008   def k \<equiv> "min 1 (r / (2 * dist x y))"
  2009   def z \<equiv> "y + scaleR k (x - y)"
  2010   have z_def2: "z = x + scaleR (1 - k) (y - x)"
  2011     unfolding z_def by (simp add: algebra_simps)
  2012   have "dist z y < r"
  2013     unfolding z_def k_def using `0 < r`
  2014     by (simp add: dist_norm min_def)
  2015   hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
  2016   have "dist x z < dist x y"
  2017     unfolding z_def2 dist_norm
  2018     apply (simp add: norm_minus_commute)
  2019     apply (simp only: dist_norm [symmetric])
  2020     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
  2021     apply (rule mult_strict_right_mono)
  2022     apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
  2023     apply (simp add: zero_less_dist_iff `x \<noteq> y`)
  2024     done
  2025   hence "z \<in> ball x (dist x y)" by simp
  2026   have "z \<noteq> y"
  2027     unfolding z_def k_def using `x \<noteq> y` `0 < r`
  2028     by (simp add: min_def)
  2029   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
  2030     using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
  2031     by fast
  2032 qed
  2033 
  2034 lemma closure_ball:
  2035   fixes x :: "'a::real_normed_vector"
  2036   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
  2037 apply (rule equalityI)
  2038 apply (rule closure_minimal)
  2039 apply (rule ball_subset_cball)
  2040 apply (rule closed_cball)
  2041 apply (rule subsetI, rename_tac y)
  2042 apply (simp add: le_less [where 'a=real])
  2043 apply (erule disjE)
  2044 apply (rule subsetD [OF closure_subset], simp)
  2045 apply (simp add: closure_def)
  2046 apply clarify
  2047 apply (rule closure_ball_lemma)
  2048 apply (simp add: zero_less_dist_iff)
  2049 done
  2050 
  2051 (* In a trivial vector space, this fails for e = 0. *)
  2052 lemma interior_cball:
  2053   fixes x :: "'a::{real_normed_vector, perfect_space}"
  2054   shows "interior (cball x e) = ball x e"
  2055 proof(cases "e\<ge>0")
  2056   case False note cs = this
  2057   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
  2058   { fix y assume "y \<in> cball x e"
  2059     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
  2060   hence "cball x e = {}" by auto
  2061   hence "interior (cball x e) = {}" using interior_empty by auto
  2062   ultimately show ?thesis by blast
  2063 next
  2064   case True note cs = this
  2065   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
  2066   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
  2067     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
  2068 
  2069     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
  2070       using perfect_choose_dist [of d] by auto
  2071     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
  2072     hence xa_cball:"xa \<in> cball x e" using as(1) by auto
  2073 
  2074     hence "y \<in> ball x e" proof(cases "x = y")
  2075       case True
  2076       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
  2077       thus "y \<in> ball x e" using `x = y ` by simp
  2078     next
  2079       case False
  2080       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
  2081         using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
  2082       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
  2083       have "y - x \<noteq> 0" using `x \<noteq> y` by auto
  2084       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
  2085         using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
  2086 
  2087       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
  2088         by (auto simp add: dist_norm algebra_simps)
  2089       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
  2090         by (auto simp add: algebra_simps)
  2091       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
  2092         using ** by auto
  2093       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)
  2094       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
  2095       thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
  2096     qed  }
  2097   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
  2098   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
  2099 qed
  2100 
  2101 lemma frontier_ball:
  2102   fixes a :: "'a::real_normed_vector"
  2103   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
  2104   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
  2105   apply (simp add: set_eq_iff)
  2106   by arith
  2107 
  2108 lemma frontier_cball:
  2109   fixes a :: "'a::{real_normed_vector, perfect_space}"
  2110   shows "frontier(cball a e) = {x. dist a x = e}"
  2111   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
  2112   apply (simp add: set_eq_iff)
  2113   by arith
  2114 
  2115 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
  2116   apply (simp add: set_eq_iff not_le)
  2117   by (metis zero_le_dist dist_self order_less_le_trans)
  2118 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
  2119 
  2120 lemma cball_eq_sing:
  2121   fixes x :: "'a::{metric_space,perfect_space}"
  2122   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
  2123 proof (rule linorder_cases)
  2124   assume e: "0 < e"
  2125   obtain a where "a \<noteq> x" "dist a x < e"
  2126     using perfect_choose_dist [OF e] by auto
  2127   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
  2128   with e show ?thesis by (auto simp add: set_eq_iff)
  2129 qed auto
  2130 
  2131 lemma cball_sing:
  2132   fixes x :: "'a::metric_space"
  2133   shows "e = 0 ==> cball x e = {x}"
  2134   by (auto simp add: set_eq_iff)
  2135 
  2136 
  2137 subsection {* Boundedness *}
  2138 
  2139   (* FIXME: This has to be unified with BSEQ!! *)
  2140 definition (in metric_space)
  2141   bounded :: "'a set \<Rightarrow> bool" where
  2142   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
  2143 
  2144 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"
  2145   unfolding bounded_def subset_eq by auto
  2146 
  2147 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
  2148 unfolding bounded_def
  2149 apply safe
  2150 apply (rule_tac x="dist a x + e" in exI, clarify)
  2151 apply (drule (1) bspec)
  2152 apply (erule order_trans [OF dist_triangle add_left_mono])
  2153 apply auto
  2154 done
  2155 
  2156 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
  2157 unfolding bounded_any_center [where a=0]
  2158 by (simp add: dist_norm)
  2159 
  2160 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
  2161   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
  2162   using assms by auto
  2163 
  2164 lemma bounded_empty [simp]: "bounded {}"
  2165   by (simp add: bounded_def)
  2166 
  2167 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
  2168   by (metis bounded_def subset_eq)
  2169 
  2170 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
  2171   by (metis bounded_subset interior_subset)
  2172 
  2173 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
  2174 proof-
  2175   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
  2176   { fix y assume "y \<in> closure S"
  2177     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
  2178       unfolding closure_sequential by auto
  2179     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
  2180     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
  2181       by (rule eventually_mono, simp add: f(1))
  2182     have "dist x y \<le> a"
  2183       apply (rule Lim_dist_ubound [of sequentially f])
  2184       apply (rule trivial_limit_sequentially)
  2185       apply (rule f(2))
  2186       apply fact
  2187       done
  2188   }
  2189   thus ?thesis unfolding bounded_def by auto
  2190 qed
  2191 
  2192 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
  2193   apply (simp add: bounded_def)
  2194   apply (rule_tac x=x in exI)
  2195   apply (rule_tac x=e in exI)
  2196   apply auto
  2197   done
  2198 
  2199 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
  2200   by (metis ball_subset_cball bounded_cball bounded_subset)
  2201 
  2202 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
  2203   apply (auto simp add: bounded_def)
  2204   apply (rename_tac x y r s)
  2205   apply (rule_tac x=x in exI)
  2206   apply (rule_tac x="max r (dist x y + s)" in exI)
  2207   apply (rule ballI, rename_tac z, safe)
  2208   apply (drule (1) bspec, simp)
  2209   apply (drule (1) bspec)
  2210   apply (rule min_max.le_supI2)
  2211   apply (erule order_trans [OF dist_triangle add_left_mono])
  2212   done
  2213 
  2214 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
  2215   by (induct rule: finite_induct[of F], auto)
  2216 
  2217 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
  2218   by (induct set: finite, auto)
  2219 
  2220 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
  2221 proof -
  2222   have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp
  2223   hence "bounded {x}" unfolding bounded_def by fast
  2224   thus ?thesis by (metis insert_is_Un bounded_Un)
  2225 qed
  2226 
  2227 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
  2228   by (induct set: finite, simp_all)
  2229 
  2230 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
  2231   apply (simp add: bounded_iff)
  2232   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
  2233   by metis arith
  2234 
  2235 lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f)"
  2236   unfolding Bseq_def bounded_pos by auto
  2237 
  2238 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
  2239   by (metis Int_lower1 Int_lower2 bounded_subset)
  2240 
  2241 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
  2242 apply (metis Diff_subset bounded_subset)
  2243 done
  2244 
  2245 lemma not_bounded_UNIV[simp, intro]:
  2246   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
  2247 proof(auto simp add: bounded_pos not_le)
  2248   obtain x :: 'a where "x \<noteq> 0"
  2249     using perfect_choose_dist [OF zero_less_one] by fast
  2250   fix b::real  assume b: "b >0"
  2251   have b1: "b +1 \<ge> 0" using b by simp
  2252   with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
  2253     by (simp add: norm_sgn)
  2254   then show "\<exists>x::'a. b < norm x" ..
  2255 qed
  2256 
  2257 lemma bounded_linear_image:
  2258   assumes "bounded S" "bounded_linear f"
  2259   shows "bounded(f ` S)"
  2260 proof-
  2261   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  2262   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)
  2263   { fix x assume "x\<in>S"
  2264     hence "norm x \<le> b" using b by auto
  2265     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
  2266       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
  2267   }
  2268   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
  2269     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
  2270 qed
  2271 
  2272 lemma bounded_scaling:
  2273   fixes S :: "'a::real_normed_vector set"
  2274   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
  2275   apply (rule bounded_linear_image, assumption)
  2276   apply (rule bounded_linear_scaleR_right)
  2277   done
  2278 
  2279 lemma bounded_translation:
  2280   fixes S :: "'a::real_normed_vector set"
  2281   assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
  2282 proof-
  2283   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
  2284   { fix x assume "x\<in>S"
  2285     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
  2286   }
  2287   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
  2288     by (auto intro!: exI[of _ "b + norm a"])
  2289 qed
  2290 
  2291 
  2292 text{* Some theorems on sups and infs using the notion "bounded". *}
  2293 
  2294 lemma bounded_real:
  2295   fixes S :: "real set"
  2296   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"
  2297   by (simp add: bounded_iff)
  2298 
  2299 lemma bounded_has_Sup:
  2300   fixes S :: "real set"
  2301   assumes "bounded S" "S \<noteq> {}"
  2302   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
  2303 proof
  2304   fix x assume "x\<in>S"
  2305   thus "x \<le> Sup S"
  2306     by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
  2307 next
  2308   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
  2309     by (metis SupInf.Sup_least)
  2310 qed
  2311 
  2312 lemma Sup_insert:
  2313   fixes S :: "real set"
  2314   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))" 
  2315 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal) 
  2316 
  2317 lemma Sup_insert_finite:
  2318   fixes S :: "real set"
  2319   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
  2320   apply (rule Sup_insert)
  2321   apply (rule finite_imp_bounded)
  2322   by simp
  2323 
  2324 lemma bounded_has_Inf:
  2325   fixes S :: "real set"
  2326   assumes "bounded S"  "S \<noteq> {}"
  2327   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
  2328 proof
  2329   fix x assume "x\<in>S"
  2330   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
  2331   thus "x \<ge> Inf S" using `x\<in>S`
  2332     by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
  2333 next
  2334   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
  2335     by (metis SupInf.Inf_greatest)
  2336 qed
  2337 
  2338 lemma Inf_insert:
  2339   fixes S :: "real set"
  2340   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))" 
  2341 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)
  2342 
  2343 lemma Inf_insert_finite:
  2344   fixes S :: "real set"
  2345   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
  2346   by (rule Inf_insert, rule finite_imp_bounded, simp)
  2347 
  2348 subsection {* Compactness *}
  2349 
  2350 subsubsection{* Open-cover compactness *}
  2351 
  2352 definition compact :: "'a::topological_space set \<Rightarrow> bool" where
  2353   compact_eq_heine_borel: -- "This name is used for backwards compatibility"
  2354     "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
  2355 
  2356 lemma compactI:
  2357   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union> C \<Longrightarrow> \<exists>C'. C' \<subseteq> C \<and> finite C' \<and> s \<subseteq> \<Union> C'"
  2358   shows "compact s"
  2359   unfolding compact_eq_heine_borel using assms by metis
  2360 
  2361 lemma compactE:
  2362   assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
  2363   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
  2364   using assms unfolding compact_eq_heine_borel by metis
  2365 
  2366 lemma compactE_image:
  2367   assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
  2368   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
  2369   using assms unfolding ball_simps[symmetric] SUP_def
  2370   by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
  2371 
  2372 subsubsection {* Bolzano-Weierstrass property *}
  2373 
  2374 lemma heine_borel_imp_bolzano_weierstrass:
  2375   assumes "compact s" "infinite t"  "t \<subseteq> s"
  2376   shows "\<exists>x \<in> s. x islimpt t"
  2377 proof(rule ccontr)
  2378   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
  2379   then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)" unfolding islimpt_def
  2380     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"] by auto
  2381   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
  2382     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
  2383   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
  2384   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
  2385     hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and `t\<subseteq>s` by auto
  2386     hence "x = y" using `f x = f y` and f[THEN bspec[where x=y]] and `y\<in>t` and `t\<subseteq>s` by auto  }
  2387   hence "inj_on f t" unfolding inj_on_def by simp
  2388   hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
  2389   moreover
  2390   { fix x assume "x\<in>t" "f x \<notin> g"
  2391     from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
  2392     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
  2393     hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
  2394     hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto  }
  2395   hence "f ` t \<subseteq> g" by auto
  2396   ultimately show False using g(2) using finite_subset by auto
  2397 qed
  2398 
  2399 lemma acc_point_range_imp_convergent_subsequence:
  2400   fixes l :: "'a :: first_countable_topology"
  2401   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
  2402   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  2403 proof -
  2404   from countable_basis_at_decseq[of l] guess A . note A = this
  2405 
  2406   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"
  2407   { fix n i
  2408     have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
  2409       using l A by auto
  2410     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
  2411       unfolding ex_in_conv by (intro notI) simp
  2412     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
  2413       by auto
  2414     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
  2415       by (auto simp: not_le)
  2416     then have "i < s n i" "f (s n i) \<in> A (Suc n)"
  2417       unfolding s_def by (auto intro: someI2_ex) }
  2418   note s = this
  2419   def r \<equiv> "nat_rec (s 0 0) s"
  2420   have "subseq r"
  2421     by (auto simp: r_def s subseq_Suc_iff)
  2422   moreover
  2423   have "(\<lambda>n. f (r n)) ----> l"
  2424   proof (rule topological_tendstoI)
  2425     fix S assume "open S" "l \<in> S"
  2426     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
  2427     moreover
  2428     { fix i assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"
  2429         by (cases i) (simp_all add: r_def s) }
  2430     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)
  2431     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
  2432       by eventually_elim auto
  2433   qed
  2434   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  2435     by (auto simp: convergent_def comp_def)
  2436 qed
  2437 
  2438 lemma sequence_infinite_lemma:
  2439   fixes f :: "nat \<Rightarrow> 'a::t1_space"
  2440   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
  2441   shows "infinite (range f)"
  2442 proof
  2443   assume "finite (range f)"
  2444   hence "closed (range f)" by (rule finite_imp_closed)
  2445   hence "open (- range f)" by (rule open_Compl)
  2446   from assms(1) have "l \<in> - range f" by auto
  2447   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
  2448     using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
  2449   thus False unfolding eventually_sequentially by auto
  2450 qed
  2451 
  2452 lemma closure_insert:
  2453   fixes x :: "'a::t1_space"
  2454   shows "closure (insert x s) = insert x (closure s)"
  2455 apply (rule closure_unique)
  2456 apply (rule insert_mono [OF closure_subset])
  2457 apply (rule closed_insert [OF closed_closure])
  2458 apply (simp add: closure_minimal)
  2459 done
  2460 
  2461 lemma islimpt_insert:
  2462   fixes x :: "'a::t1_space"
  2463   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
  2464 proof
  2465   assume *: "x islimpt (insert a s)"
  2466   show "x islimpt s"
  2467   proof (rule islimptI)
  2468     fix t assume t: "x \<in> t" "open t"
  2469     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
  2470     proof (cases "x = a")
  2471       case True
  2472       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
  2473         using * t by (rule islimptE)
  2474       with `x = a` show ?thesis by auto
  2475     next
  2476       case False
  2477       with t have t': "x \<in> t - {a}" "open (t - {a})"
  2478         by (simp_all add: open_Diff)
  2479       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
  2480         using * t' by (rule islimptE)
  2481       thus ?thesis by auto
  2482     qed
  2483   qed
  2484 next
  2485   assume "x islimpt s" thus "x islimpt (insert a s)"
  2486     by (rule islimpt_subset) auto
  2487 qed
  2488 
  2489 lemma islimpt_finite:
  2490   fixes x :: "'a::t1_space"
  2491   shows "finite s \<Longrightarrow> \<not> x islimpt s"
  2492 by (induct set: finite, simp_all add: islimpt_insert)
  2493 
  2494 lemma islimpt_union_finite:
  2495   fixes x :: "'a::t1_space"
  2496   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
  2497 by (simp add: islimpt_Un islimpt_finite)
  2498 
  2499 lemma islimpt_eq_acc_point:
  2500   fixes l :: "'a :: t1_space"
  2501   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
  2502 proof (safe intro!: islimptI)
  2503   fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
  2504   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
  2505     by (auto intro: finite_imp_closed)
  2506   then show False
  2507     by (rule islimptE) auto
  2508 next
  2509   fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
  2510   then have "infinite (T \<inter> S - {l})" by auto
  2511   then have "\<exists>x. x \<in> (T \<inter> S - {l})"
  2512     unfolding ex_in_conv by (intro notI) simp
  2513   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
  2514     by auto
  2515 qed
  2516 
  2517 lemma islimpt_range_imp_convergent_subsequence:
  2518   fixes l :: "'a :: {t1_space, first_countable_topology}"
  2519   assumes l: "l islimpt (range f)"
  2520   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  2521   using l unfolding islimpt_eq_acc_point
  2522   by (rule acc_point_range_imp_convergent_subsequence)
  2523 
  2524 lemma sequence_unique_limpt:
  2525   fixes f :: "nat \<Rightarrow> 'a::t2_space"
  2526   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
  2527   shows "l' = l"
  2528 proof (rule ccontr)
  2529   assume "l' \<noteq> l"
  2530   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
  2531     using hausdorff [OF `l' \<noteq> l`] by auto
  2532   have "eventually (\<lambda>n. f n \<in> t) sequentially"
  2533     using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
  2534   then obtain N where "\<forall>n\<ge>N. f n \<in> t"
  2535     unfolding eventually_sequentially by auto
  2536 
  2537   have "UNIV = {..<N} \<union> {N..}" by auto
  2538   hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
  2539   hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
  2540   hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
  2541   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
  2542     using `l' \<in> s` `open s` by (rule islimptE)
  2543   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
  2544   with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
  2545   with `s \<inter> t = {}` show False by simp
  2546 qed
  2547 
  2548 lemma bolzano_weierstrass_imp_closed:
  2549   fixes s :: "'a::{first_countable_topology, t2_space} set"
  2550   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
  2551   shows "closed s"
  2552 proof-
  2553   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
  2554     hence "l \<in> s"
  2555     proof(cases "\<forall>n. x n \<noteq> l")
  2556       case False thus "l\<in>s" using as(1) by auto
  2557     next
  2558       case True note cas = this
  2559       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
  2560       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
  2561       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
  2562     qed  }
  2563   thus ?thesis unfolding closed_sequential_limits by fast
  2564 qed
  2565 
  2566 lemma compact_imp_closed:
  2567   fixes s :: "'a::t2_space set"
  2568   assumes "compact s" shows "closed s"
  2569 unfolding closed_def
  2570 proof (rule openI)
  2571   fix y assume "y \<in> - s"
  2572   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
  2573   note `compact s`
  2574   moreover have "\<forall>u\<in>?C. open u" by simp
  2575   moreover have "s \<subseteq> \<Union>?C"
  2576   proof
  2577     fix x assume "x \<in> s"
  2578     with `y \<in> - s` have "x \<noteq> y" by clarsimp
  2579     hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
  2580       by (rule hausdorff)
  2581     with `x \<in> s` show "x \<in> \<Union>?C"
  2582       unfolding eventually_nhds by auto
  2583   qed
  2584   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
  2585     by (rule compactE)
  2586   from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
  2587   with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
  2588     by (simp add: eventually_Ball_finite)
  2589   with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
  2590     by (auto elim!: eventually_mono [rotated])
  2591   thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
  2592     by (simp add: eventually_nhds subset_eq)
  2593 qed
  2594 
  2595 lemma compact_imp_bounded:
  2596   assumes "compact U" shows "bounded U"
  2597 proof -
  2598   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)" using assms by auto
  2599   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
  2600     by (elim compactE_image)
  2601   from `finite D` have "bounded (\<Union>x\<in>D. ball x 1)"
  2602     by (simp add: bounded_UN)
  2603   thus "bounded U" using `U \<subseteq> (\<Union>x\<in>D. ball x 1)` 
  2604     by (rule bounded_subset)
  2605 qed
  2606 
  2607 text{* In particular, some common special cases. *}
  2608 
  2609 lemma compact_empty[simp]:
  2610  "compact {}"
  2611   unfolding compact_eq_heine_borel
  2612   by auto
  2613 
  2614 lemma compact_union [intro]:
  2615   assumes "compact s" "compact t" shows " compact (s \<union> t)"
  2616 proof (rule compactI)
  2617   fix f assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
  2618   from * `compact s` obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
  2619     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis
  2620   moreover from * `compact t` obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
  2621     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis
  2622   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
  2623     by (auto intro!: exI[of _ "s' \<union> t'"])
  2624 qed
  2625 
  2626 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
  2627   by (induct set: finite) auto
  2628 
  2629 lemma compact_UN [intro]:
  2630   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
  2631   unfolding SUP_def by (rule compact_Union) auto
  2632 
  2633 lemma compact_inter_closed [intro]:
  2634   assumes "compact s" and "closed t"
  2635   shows "compact (s \<inter> t)"
  2636 proof (rule compactI)
  2637   fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
  2638   from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto
  2639   moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
  2640   ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
  2641     using `compact s` unfolding compact_eq_heine_borel by auto
  2642   then guess D ..
  2643   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
  2644     by (intro exI[of _ "D - {-t}"]) auto
  2645 qed
  2646 
  2647 lemma closed_inter_compact [intro]:
  2648   assumes "closed s" and "compact t"
  2649   shows "compact (s \<inter> t)"
  2650   using compact_inter_closed [of t s] assms
  2651   by (simp add: Int_commute)
  2652 
  2653 lemma compact_inter [intro]:
  2654   fixes s t :: "'a :: t2_space set"
  2655   assumes "compact s" and "compact t"
  2656   shows "compact (s \<inter> t)"
  2657   using assms by (intro compact_inter_closed compact_imp_closed)
  2658 
  2659 lemma compact_sing [simp]: "compact {a}"
  2660   unfolding compact_eq_heine_borel by auto
  2661 
  2662 lemma compact_insert [simp]:
  2663   assumes "compact s" shows "compact (insert x s)"
  2664 proof -
  2665   have "compact ({x} \<union> s)"
  2666     using compact_sing assms by (rule compact_union)
  2667   thus ?thesis by simp
  2668 qed
  2669 
  2670 lemma finite_imp_compact:
  2671   shows "finite s \<Longrightarrow> compact s"
  2672   by (induct set: finite) simp_all
  2673 
  2674 lemma open_delete:
  2675   fixes s :: "'a::t1_space set"
  2676   shows "open s \<Longrightarrow> open (s - {x})"
  2677   by (simp add: open_Diff)
  2678 
  2679 text{* Finite intersection property *}
  2680 
  2681 lemma inj_setminus: "inj_on uminus (A::'a set set)"
  2682   by (auto simp: inj_on_def)
  2683 
  2684 lemma compact_fip:
  2685   "compact U \<longleftrightarrow>
  2686     (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"
  2687   (is "_ \<longleftrightarrow> ?R")
  2688 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
  2689   fix A assume "compact U" and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
  2690     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
  2691   from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>uminus`A"
  2692     by auto
  2693   with `compact U` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
  2694     unfolding compact_eq_heine_borel by (metis subset_image_iff)
  2695   with fi[THEN spec, of B] show False
  2696     by (auto dest: finite_imageD intro: inj_setminus)
  2697 next
  2698   fix A assume ?R and cover: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
  2699   from cover have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
  2700     by auto
  2701   with `?R` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>uminus`B = {}"
  2702     by (metis subset_image_iff)
  2703   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
  2704     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
  2705 qed
  2706 
  2707 lemma compact_imp_fip:
  2708   "compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>
  2709     s \<inter> (\<Inter> f) \<noteq> {}"
  2710   unfolding compact_fip by auto
  2711 
  2712 text{*Compactness expressed with filters*}
  2713 
  2714 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"
  2715 
  2716 lemma eventually_filter_from_subbase:
  2717   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"
  2718     (is "_ \<longleftrightarrow> ?R P")
  2719   unfolding filter_from_subbase_def
  2720 proof (rule eventually_Abs_filter is_filter.intro)+
  2721   show "?R (\<lambda>x. True)"
  2722     by (rule exI[of _ "{}"]) (simp add: le_fun_def)
  2723 next
  2724   fix P Q assume "?R P" then guess X ..
  2725   moreover assume "?R Q" then guess Y ..
  2726   ultimately show "?R (\<lambda>x. P x \<and> Q x)"
  2727     by (intro exI[of _ "X \<union> Y"]) auto
  2728 next
  2729   fix P Q
  2730   assume "?R P" then guess X ..
  2731   moreover assume "\<forall>x. P x \<longrightarrow> Q x"
  2732   ultimately show "?R Q"
  2733     by (intro exI[of _ X]) auto
  2734 qed
  2735 
  2736 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"
  2737   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])
  2738 
  2739 lemma filter_from_subbase_not_bot:
  2740   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"
  2741   unfolding trivial_limit_def eventually_filter_from_subbase by auto
  2742 
  2743 lemma closure_iff_nhds_not_empty:
  2744   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
  2745 proof safe
  2746   assume x: "x \<in> closure X"
  2747   fix S A assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
  2748   then have "x \<notin> closure (-S)" 
  2749     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
  2750   with x have "x \<in> closure X - closure (-S)"
  2751     by auto
  2752   also have "\<dots> \<subseteq> closure (X \<inter> S)"
  2753     using `open S` open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
  2754   finally have "X \<inter> S \<noteq> {}" by auto
  2755   then show False using `X \<inter> A = {}` `S \<subseteq> A` by auto
  2756 next
  2757   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
  2758   from this[THEN spec, of "- X", THEN spec, of "- closure X"]
  2759   show "x \<in> closure X"
  2760     by (simp add: closure_subset open_Compl)
  2761 qed
  2762 
  2763 lemma compact_filter:
  2764   "compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
  2765 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
  2766   fix F assume "compact U" and F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
  2767   from F have "U \<noteq> {}"
  2768     by (auto simp: eventually_False)
  2769 
  2770   def Z \<equiv> "closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
  2771   then have "\<forall>z\<in>Z. closed z"
  2772     by auto
  2773   moreover 
  2774   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
  2775     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])
  2776   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
  2777   proof (intro allI impI)
  2778     fix B assume "finite B" "B \<subseteq> Z"
  2779     with `finite B` ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"
  2780       by (auto intro!: eventually_Ball_finite)
  2781     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
  2782       by eventually_elim auto
  2783     with F show "U \<inter> \<Inter>B \<noteq> {}"
  2784       by (intro notI) (simp add: eventually_False)
  2785   qed
  2786   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
  2787     using `compact U` unfolding compact_fip by blast
  2788   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z" by auto
  2789 
  2790   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
  2791     unfolding eventually_inf eventually_nhds
  2792   proof safe
  2793     fix P Q R S
  2794     assume "eventually R F" "open S" "x \<in> S"
  2795     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
  2796     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
  2797     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
  2798     ultimately show False by (auto simp: set_eq_iff)
  2799   qed
  2800   with `x \<in> U` show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
  2801     by (metis eventually_bot)
  2802 next
  2803   fix A assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
  2804 
  2805   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"
  2806   then have inj_P': "\<And>A. inj_on P' A"
  2807     by (auto intro!: inj_onI simp: fun_eq_iff)
  2808   def F \<equiv> "filter_from_subbase (P' ` insert U A)"
  2809   have "F \<noteq> bot"
  2810     unfolding F_def
  2811   proof (safe intro!: filter_from_subbase_not_bot)
  2812     fix X assume "X \<subseteq> P' ` insert U A" "finite X" "Inf X = bot"
  2813     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P' ` B) = bot"
  2814       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)
  2815     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}" by auto
  2816     with B show False by (auto simp: P'_def fun_eq_iff)
  2817   qed
  2818   moreover have "eventually (\<lambda>x. x \<in> U) F"
  2819     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)
  2820   moreover assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
  2821   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
  2822     by auto
  2823 
  2824   { fix V assume "V \<in> A"
  2825     then have V: "eventually (\<lambda>x. x \<in> V) F"
  2826       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)
  2827     have "x \<in> closure V"
  2828       unfolding closure_iff_nhds_not_empty
  2829     proof (intro impI allI)
  2830       fix S A assume "open S" "x \<in> S" "S \<subseteq> A"
  2831       then have "eventually (\<lambda>x. x \<in> A) (nhds x)" by (auto simp: eventually_nhds)
  2832       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
  2833         by (auto simp: eventually_inf)
  2834       with x show "V \<inter> A \<noteq> {}"
  2835         by (auto simp del: Int_iff simp add: trivial_limit_def)
  2836     qed
  2837     then have "x \<in> V"
  2838       using `V \<in> A` A(1) by simp }
  2839   with `x\<in>U` have "x \<in> U \<inter> \<Inter>A" by auto
  2840   with `U \<inter> \<Inter>A = {}` show False by auto
  2841 qed
  2842 
  2843 definition "countably_compact U \<longleftrightarrow>
  2844     (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"
  2845 
  2846 lemma countably_compactE:
  2847   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
  2848   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
  2849   using assms unfolding countably_compact_def by metis
  2850 
  2851 lemma countably_compactI:
  2852   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"
  2853   shows "countably_compact s"
  2854   using assms unfolding countably_compact_def by metis
  2855 
  2856 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
  2857   by (auto simp: compact_eq_heine_borel countably_compact_def)
  2858 
  2859 lemma countably_compact_imp_compact:
  2860   assumes "countably_compact U"
  2861   assumes ccover: "countable B" "\<forall>b\<in>B. open b"
  2862   assumes basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
  2863   shows "compact U"
  2864   using `countably_compact U` unfolding compact_eq_heine_borel countably_compact_def
  2865 proof safe
  2866   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
  2867   assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
  2868 
  2869   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
  2870   ultimately have "countable C" "\<forall>a\<in>C. open a"
  2871     unfolding C_def using ccover by auto
  2872   moreover
  2873   have "\<Union>A \<inter> U \<subseteq> \<Union>C"
  2874   proof safe
  2875     fix x a assume "x \<in> U" "x \<in> a" "a \<in> A"
  2876     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a" by blast
  2877     with `a \<in> A` show "x \<in> \<Union>C" unfolding C_def
  2878       by auto
  2879   qed
  2880   then have "U \<subseteq> \<Union>C" using `U \<subseteq> \<Union>A` by auto
  2881   ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
  2882     using * by metis
  2883   moreover then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
  2884     by (auto simp: C_def)
  2885   then guess f unfolding bchoice_iff Bex_def ..
  2886   ultimately show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
  2887     unfolding C_def by (intro exI[of _ "f`T"]) fastforce
  2888 qed
  2889 
  2890 lemma countably_compact_imp_compact_second_countable:
  2891   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
  2892 proof (rule countably_compact_imp_compact)
  2893   fix T and x :: 'a assume "open T" "x \<in> T"
  2894   from topological_basisE[OF is_basis this] guess b .
  2895   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T" by auto
  2896 qed (insert countable_basis topological_basis_open[OF is_basis], auto)
  2897 
  2898 lemma countably_compact_eq_compact:
  2899   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
  2900   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
  2901   
  2902 subsubsection{* Sequential compactness *}
  2903 
  2904 definition
  2905   seq_compact :: "'a::topological_space set \<Rightarrow> bool" where
  2906   "seq_compact S \<longleftrightarrow>
  2907    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
  2908        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
  2909 
  2910 lemma seq_compact_imp_countably_compact:
  2911   fixes U :: "'a :: first_countable_topology set"
  2912   assumes "seq_compact U"
  2913   shows "countably_compact U"
  2914 proof (safe intro!: countably_compactI)
  2915   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
  2916   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"
  2917     using `seq_compact U` by (fastforce simp: seq_compact_def subset_eq)
  2918   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
  2919   proof cases
  2920     assume "finite A" with A show ?thesis by auto
  2921   next
  2922     assume "infinite A"
  2923     then have "A \<noteq> {}" by auto
  2924     show ?thesis
  2925     proof (rule ccontr)
  2926       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
  2927       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)" by auto
  2928       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T" by metis
  2929       def X \<equiv> "\<lambda>n. X' (from_nat_into A ` {.. n})"
  2930       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
  2931         using `A \<noteq> {}` unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)
  2932       then have "range X \<subseteq> U" by auto
  2933       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x" by auto
  2934       from `x\<in>U` `U \<subseteq> \<Union>A` from_nat_into_surj[OF `countable A`]
  2935       obtain n where "x \<in> from_nat_into A n" by auto
  2936       with r(2) A(1) from_nat_into[OF `A \<noteq> {}`, of n]
  2937       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
  2938         unfolding tendsto_def by (auto simp: comp_def)
  2939       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
  2940         by (auto simp: eventually_sequentially)
  2941       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
  2942         by auto
  2943       moreover from `subseq r`[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
  2944         by (auto intro!: exI[of _ "max n N"])
  2945       ultimately show False
  2946         by auto
  2947     qed
  2948   qed
  2949 qed
  2950 
  2951 lemma compact_imp_seq_compact:
  2952   fixes U :: "'a :: first_countable_topology set"
  2953   assumes "compact U" shows "seq_compact U"
  2954   unfolding seq_compact_def
  2955 proof safe
  2956   fix X :: "nat \<Rightarrow> 'a" assume "\<forall>n. X n \<in> U"
  2957   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
  2958     by (auto simp: eventually_filtermap)
  2959   moreover have "filtermap X sequentially \<noteq> bot"
  2960     by (simp add: trivial_limit_def eventually_filtermap)
  2961   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
  2962     using `compact U` by (auto simp: compact_filter)
  2963 
  2964   from countable_basis_at_decseq[of x] guess A . note A = this
  2965   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"
  2966   { fix n i
  2967     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
  2968     proof (rule ccontr)
  2969       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
  2970       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)" by auto
  2971       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
  2972         by (auto simp: eventually_filtermap eventually_sequentially)
  2973       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
  2974         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
  2975       ultimately have "eventually (\<lambda>x. False) ?F"
  2976         by (auto simp add: eventually_inf)
  2977       with x show False
  2978         by (simp add: eventually_False)
  2979     qed
  2980     then have "i < s n i" "X (s n i) \<in> A (Suc n)"
  2981       unfolding s_def by (auto intro: someI2_ex) }
  2982   note s = this
  2983   def r \<equiv> "nat_rec (s 0 0) s"
  2984   have "subseq r"
  2985     by (auto simp: r_def s subseq_Suc_iff)
  2986   moreover
  2987   have "(\<lambda>n. X (r n)) ----> x"
  2988   proof (rule topological_tendstoI)
  2989     fix S assume "open S" "x \<in> S"
  2990     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
  2991     moreover
  2992     { fix i assume "Suc 0 \<le> i" then have "X (r i) \<in> A i"
  2993         by (cases i) (simp_all add: r_def s) }
  2994     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)
  2995     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
  2996       by eventually_elim auto
  2997   qed
  2998   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"
  2999     using `x \<in> U` by (auto simp: convergent_def comp_def)
  3000 qed
  3001 
  3002 lemma seq_compactI:
  3003   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
  3004   shows "seq_compact S"
  3005   unfolding seq_compact_def using assms by fast
  3006 
  3007 lemma seq_compactE:
  3008   assumes "seq_compact S" "\<forall>n. f n \<in> S"
  3009   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
  3010   using assms unfolding seq_compact_def by fast
  3011 
  3012 lemma countably_compact_imp_acc_point:
  3013   assumes "countably_compact s" "countable t" "infinite t"  "t \<subseteq> s"
  3014   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
  3015 proof (rule ccontr)
  3016   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"  
  3017   note `countably_compact s`
  3018   moreover have "\<forall>t\<in>C. open t" 
  3019     by (auto simp: C_def)
  3020   moreover
  3021   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
  3022   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
  3023   have "s \<subseteq> \<Union>C"
  3024     using `t \<subseteq> s`
  3025     unfolding C_def Union_image_eq
  3026     apply (safe dest!: s)
  3027     apply (rule_tac a="U \<inter> t" in UN_I)
  3028     apply (auto intro!: interiorI simp add: finite_subset)
  3029     done
  3030   moreover
  3031   from `countable t` have "countable C"
  3032     unfolding C_def by (auto intro: countable_Collect_finite_subset)
  3033   ultimately guess D by (rule countably_compactE)
  3034   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E" and
  3035     s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
  3036     by (metis (lifting) Union_image_eq finite_subset_image C_def)
  3037   from s `t \<subseteq> s` have "t \<subseteq> \<Union>E"
  3038     using interior_subset by blast
  3039   moreover have "finite (\<Union>E)"
  3040     using E by auto
  3041   ultimately show False using `infinite t` by (auto simp: finite_subset)
  3042 qed
  3043 
  3044 lemma countable_acc_point_imp_seq_compact:
  3045   fixes s :: "'a::first_countable_topology set"
  3046   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
  3047   shows "seq_compact s"
  3048 proof -
  3049   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  3050     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3051     proof (cases "finite (range f)")
  3052       case True
  3053       obtain l where "infinite {n. f n = f l}"
  3054         using pigeonhole_infinite[OF _ True] by auto
  3055       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"
  3056         using infinite_enumerate by blast
  3057       hence "subseq r \<and> (f \<circ> r) ----> f l"
  3058         by (simp add: fr tendsto_const o_def)
  3059       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  3060         by auto
  3061     next
  3062       case False
  3063       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" by auto
  3064       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
  3065       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3066         using acc_point_range_imp_convergent_subsequence[of l f] by auto
  3067       with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
  3068     qed
  3069   }
  3070   thus ?thesis unfolding seq_compact_def by auto
  3071 qed
  3072 
  3073 lemma seq_compact_eq_countably_compact:
  3074   fixes U :: "'a :: first_countable_topology set"
  3075   shows "seq_compact U \<longleftrightarrow> countably_compact U"
  3076   using
  3077     countable_acc_point_imp_seq_compact
  3078     countably_compact_imp_acc_point
  3079     seq_compact_imp_countably_compact
  3080   by metis
  3081 
  3082 lemma seq_compact_eq_acc_point:
  3083   fixes s :: "'a :: first_countable_topology set"
  3084   shows "seq_compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
  3085   using
  3086     countable_acc_point_imp_seq_compact[of s]
  3087     countably_compact_imp_acc_point[of s]
  3088     seq_compact_imp_countably_compact[of s]
  3089   by metis
  3090 
  3091 lemma seq_compact_eq_compact:
  3092   fixes U :: "'a :: second_countable_topology set"
  3093   shows "seq_compact U \<longleftrightarrow> compact U"
  3094   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
  3095 
  3096 lemma bolzano_weierstrass_imp_seq_compact:
  3097   fixes s :: "'a::{t1_space, first_countable_topology} set"
  3098   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
  3099   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
  3100 
  3101 subsubsection{* Total boundedness *}
  3102 
  3103 lemma cauchy_def:
  3104   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
  3105 unfolding Cauchy_def by blast
  3106 
  3107 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
  3108   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
  3109 declare helper_1.simps[simp del]
  3110 
  3111 lemma seq_compact_imp_totally_bounded:
  3112   assumes "seq_compact s"
  3113   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
  3114 proof(rule, rule, rule ccontr)
  3115   fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k)"
  3116   def x \<equiv> "helper_1 s e"
  3117   { fix n
  3118     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
  3119     proof(induct_tac rule:nat_less_induct)
  3120       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
  3121       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
  3122       have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x ` {0 ..< n}" in allE) using as by auto
  3123       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
  3124       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
  3125         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
  3126       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
  3127     qed }
  3128   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
  3129   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto
  3130   from this(3) have "Cauchy (x \<circ> r)" using LIMSEQ_imp_Cauchy by auto
  3131   then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using `e>0` by auto
  3132   show False
  3133     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
  3134     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
  3135     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
  3136 qed
  3137 
  3138 subsubsection{* Heine-Borel theorem *}
  3139 
  3140 lemma seq_compact_imp_heine_borel:
  3141   fixes s :: "'a :: metric_space set"
  3142   assumes "seq_compact s" shows "compact s"
  3143 proof -
  3144   from seq_compact_imp_totally_bounded[OF `seq_compact s`]
  3145   guess f unfolding choice_iff' .. note f = this
  3146   def K \<equiv> "(\<lambda>(x, r). ball x r) ` ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
  3147   have "countably_compact s"
  3148     using `seq_compact s` by (rule seq_compact_imp_countably_compact)
  3149   then show "compact s"
  3150   proof (rule countably_compact_imp_compact)
  3151     show "countable K"
  3152       unfolding K_def using f
  3153       by (auto intro: countable_finite countable_subset countable_rat
  3154                intro!: countable_image countable_SIGMA countable_UN)
  3155     show "\<forall>b\<in>K. open b" by (auto simp: K_def)
  3156   next
  3157     fix T x assume T: "open T" "x \<in> T" and x: "x \<in> s"
  3158     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T" by auto
  3159     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T" by auto
  3160     from Rats_dense_in_real[OF `0 < e / 2`] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2" by auto
  3161     from f[rule_format, of r] `0 < r` `x \<in> s` obtain k where "k \<in> f r" "x \<in> ball k r"
  3162       unfolding Union_image_eq by auto
  3163     from `r \<in> \<rat>` `0 < r` `k \<in> f r` have "ball k r \<in> K" by (auto simp: K_def)
  3164     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
  3165     proof (rule bexI[rotated], safe)
  3166       fix y assume "y \<in> ball k r"
  3167       with `r < e / 2` `x \<in> ball k r` have "dist x y < e"
  3168         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)
  3169       with `ball x e \<subseteq> T` show "y \<in> T" by auto
  3170     qed (rule `x \<in> ball k r`)
  3171   qed
  3172 qed
  3173 
  3174 lemma compact_eq_seq_compact_metric:
  3175   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
  3176   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
  3177 
  3178 lemma compact_def:
  3179   "compact (S :: 'a::metric_space set) \<longleftrightarrow>
  3180    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"
  3181   unfolding compact_eq_seq_compact_metric seq_compact_def by auto
  3182 
  3183 subsubsection {* Complete the chain of compactness variants *}
  3184 
  3185 lemma compact_eq_bolzano_weierstrass:
  3186   fixes s :: "'a::metric_space set"
  3187   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
  3188 proof
  3189   assume ?lhs thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto
  3190 next
  3191   assume ?rhs thus ?lhs
  3192     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
  3193 qed
  3194 
  3195 lemma bolzano_weierstrass_imp_bounded:
  3196   "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
  3197   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .
  3198 
  3199 text {*
  3200   A metric space (or topological vector space) is said to have the
  3201   Heine-Borel property if every closed and bounded subset is compact.
  3202 *}
  3203 
  3204 class heine_borel = metric_space +
  3205   assumes bounded_imp_convergent_subsequence:
  3206     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3207 
  3208 lemma bounded_closed_imp_seq_compact:
  3209   fixes s::"'a::heine_borel set"
  3210   assumes "bounded s" and "closed s" shows "seq_compact s"
  3211 proof (unfold seq_compact_def, clarify)
  3212   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  3213   with `bounded s` have "bounded (range f)" by (auto intro: bounded_subset)
  3214   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
  3215     using bounded_imp_convergent_subsequence [OF `bounded (range f)`] by auto
  3216   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
  3217   have "l \<in> s" using `closed s` fr l
  3218     unfolding closed_sequential_limits by blast
  3219   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3220     using `l \<in> s` r l by blast
  3221 qed
  3222 
  3223 lemma compact_eq_bounded_closed:
  3224   fixes s :: "'a::heine_borel set"
  3225   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")
  3226 proof
  3227   assume ?lhs thus ?rhs
  3228     using compact_imp_closed compact_imp_bounded by blast
  3229 next
  3230   assume ?rhs thus ?lhs
  3231     using bounded_closed_imp_seq_compact[of s] unfolding compact_eq_seq_compact_metric by auto
  3232 qed
  3233 
  3234 (* TODO: is this lemma necessary? *)
  3235 lemma bounded_increasing_convergent:
  3236   fixes s :: "nat \<Rightarrow> real"
  3237   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"
  3238   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]
  3239   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)
  3240 
  3241 instance real :: heine_borel
  3242 proof
  3243   fix f :: "nat \<Rightarrow> real"
  3244   assume f: "bounded (range f)"
  3245   obtain r where r: "subseq r" "monoseq (f \<circ> r)"
  3246     unfolding comp_def by (metis seq_monosub)
  3247   moreover
  3248   then have "Bseq (f \<circ> r)"
  3249     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)
  3250   ultimately show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"
  3251     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
  3252 qed
  3253 
  3254 lemma compact_lemma:
  3255   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
  3256   assumes "bounded (range f)"
  3257   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>
  3258         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
  3259 proof safe
  3260   fix d :: "'a set" assume d: "d \<subseteq> Basis" 
  3261   with finite_Basis have "finite d" by (blast intro: finite_subset)
  3262   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>
  3263       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
  3264   proof(induct d) case empty thus ?case unfolding subseq_def by auto
  3265   next case (insert k d) have k[intro]:"k\<in>Basis" using insert by auto
  3266     have s': "bounded ((\<lambda>x. x \<bullet> k) ` range f)" using `bounded (range f)`
  3267       by (auto intro!: bounded_linear_image bounded_linear_inner_left)
  3268     obtain l1::"'a" and r1 where r1:"subseq r1" and
  3269       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
  3270       using insert(3) using insert(4) by auto
  3271     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k) ` range f" by simp
  3272     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"
  3273       by (metis (lifting) bounded_subset f' image_subsetI s')
  3274     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"
  3275       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"] by (auto simp: o_def)
  3276     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
  3277       using r1 and r2 unfolding r_def o_def subseq_def by auto
  3278     moreover
  3279     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"
  3280     { fix e::real assume "e>0"
  3281       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially" by blast
  3282       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially" by (rule tendstoD)
  3283       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
  3284         by (rule eventually_subseq)
  3285       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
  3286         using N1' N2 
  3287         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)
  3288     }
  3289     ultimately show ?case by auto
  3290   qed
  3291 qed
  3292 
  3293 instance euclidean_space \<subseteq> heine_borel
  3294 proof
  3295   fix f :: "nat \<Rightarrow> 'a"
  3296   assume f: "bounded (range f)"
  3297   then obtain l::'a and r where r: "subseq r"
  3298     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
  3299     using compact_lemma [OF f] by blast
  3300   { fix e::real assume "e>0"
  3301     hence "0 < e / real_of_nat DIM('a)" by (auto intro!: divide_pos_pos DIM_positive)
  3302     with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))) sequentially"
  3303       by simp
  3304     moreover
  3305     { fix n assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"
  3306       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"
  3307         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
  3308       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"
  3309         apply(rule setsum_strict_mono) using n by auto
  3310       finally have "dist (f (r n)) l < e" 
  3311         by auto
  3312     }
  3313     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
  3314       by (rule eventually_elim1)
  3315   }
  3316   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
  3317   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
  3318 qed
  3319 
  3320 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
  3321 unfolding bounded_def
  3322 apply clarify
  3323 apply (rule_tac x="a" in exI)
  3324 apply (rule_tac x="e" in exI)
  3325 apply clarsimp
  3326 apply (drule (1) bspec)
  3327 apply (simp add: dist_Pair_Pair)
  3328 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
  3329 done
  3330 
  3331 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
  3332 unfolding bounded_def
  3333 apply clarify
  3334 apply (rule_tac x="b" in exI)
  3335 apply (rule_tac x="e" in exI)
  3336 apply clarsimp
  3337 apply (drule (1) bspec)
  3338 apply (simp add: dist_Pair_Pair)
  3339 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
  3340 done
  3341 
  3342 instance prod :: (heine_borel, heine_borel) heine_borel
  3343 proof
  3344   fix f :: "nat \<Rightarrow> 'a \<times> 'b"
  3345   assume f: "bounded (range f)"
  3346   from f have s1: "bounded (range (fst \<circ> f))" unfolding image_comp by (rule bounded_fst)
  3347   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"
  3348     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
  3349   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
  3350     by (auto simp add: image_comp intro: bounded_snd bounded_subset)
  3351   obtain l2 r2 where r2: "subseq r2"
  3352     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
  3353     using bounded_imp_convergent_subsequence [OF s2]
  3354     unfolding o_def by fast
  3355   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
  3356     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
  3357   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
  3358     using tendsto_Pair [OF l1' l2] unfolding o_def by simp
  3359   have r: "subseq (r1 \<circ> r2)"
  3360     using r1 r2 unfolding subseq_def by simp
  3361   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
  3362     using l r by fast
  3363 qed
  3364 
  3365 subsubsection{* Completeness *}
  3366 
  3367 definition complete :: "'a::metric_space set \<Rightarrow> bool" where
  3368   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"
  3369 
  3370 lemma compact_imp_complete: assumes "compact s" shows "complete s"
  3371 proof-
  3372   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
  3373     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"
  3374       using assms unfolding compact_def by blast
  3375 
  3376     note lr' = seq_suble [OF lr(2)]
  3377 
  3378     { fix e::real assume "e>0"
  3379       from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using `e>0` apply (erule_tac x="e/2" in allE) by auto
  3380       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using `e>0` by auto
  3381       { fix n::nat assume n:"n \<ge> max N M"
  3382         have "dist ((f \<circ> r) n) l < e/2" using n M by auto
  3383         moreover have "r n \<ge> N" using lr'[of n] n by auto
  3384         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
  3385         ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)  }
  3386       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }
  3387     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding LIMSEQ_def by auto  }
  3388   thus ?thesis unfolding complete_def by auto
  3389 qed
  3390 
  3391 lemma nat_approx_posE:
  3392   fixes e::real
  3393   assumes "0 < e"
  3394   obtains n::nat where "1 / (Suc n) < e"
  3395 proof atomize_elim
  3396   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"
  3397     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: `0 < e`)
  3398   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"
  3399     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: `0 < e`)
  3400   also have "\<dots> = e" by simp
  3401   finally show  "\<exists>n. 1 / real (Suc n) < e" ..
  3402 qed
  3403 
  3404 lemma compact_eq_totally_bounded:
  3405   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k)))"
  3406     (is "_ \<longleftrightarrow> ?rhs")
  3407 proof
  3408   assume assms: "?rhs"
  3409   then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> s \<subseteq> (\<Union>x\<in>k e. ball x e)"
  3410     by (auto simp: choice_iff')
  3411 
  3412   show "compact s"
  3413   proof cases
  3414     assume "s = {}" thus "compact s" by (simp add: compact_def)
  3415   next
  3416     assume "s \<noteq> {}"
  3417     show ?thesis
  3418       unfolding compact_def
  3419     proof safe
  3420       fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
  3421       
  3422       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"
  3423       then have [simp]: "\<And>n. 0 < e n" by auto
  3424       def B \<equiv> "\<lambda>n U. SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
  3425       { fix n U assume "infinite {n. f n \<in> U}"
  3426         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
  3427           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
  3428         then guess a ..
  3429         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
  3430           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
  3431         from someI_ex[OF this]
  3432         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
  3433           unfolding B_def by auto }
  3434       note B = this
  3435 
  3436       def F \<equiv> "nat_rec (B 0 UNIV) B"
  3437       { fix n have "infinite {i. f i \<in> F n}"
  3438           by (induct n) (auto simp: F_def B) }
  3439       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
  3440         using B by (simp add: F_def)
  3441       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
  3442         using decseq_SucI[of F] by (auto simp: decseq_def)
  3443 
  3444       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
  3445       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
  3446         fix k i
  3447         have "infinite ({n. f n \<in> F k} - {.. i})"
  3448           using `infinite {n. f n \<in> F k}` by auto
  3449         from infinite_imp_nonempty[OF this]
  3450         show "\<exists>x>i. f x \<in> F k"
  3451           by (simp add: set_eq_iff not_le conj_commute)
  3452       qed
  3453 
  3454       def t \<equiv> "nat_rec (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
  3455       have "subseq t"
  3456         unfolding subseq_Suc_iff by (simp add: t_def sel)
  3457       moreover have "\<forall>i. (f \<circ> t) i \<in> s"
  3458         using f by auto
  3459       moreover
  3460       { fix n have "(f \<circ> t) n \<in> F n"
  3461           by (cases n) (simp_all add: t_def sel) }
  3462       note t = this
  3463 
  3464       have "Cauchy (f \<circ> t)"
  3465       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
  3466         fix r :: real and N n m assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
  3467         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
  3468           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)
  3469         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
  3470           by (auto simp: subset_eq)
  3471         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] `2 * e N < r`
  3472         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
  3473           by (simp add: dist_commute)
  3474       qed
  3475 
  3476       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
  3477         using assms unfolding complete_def by blast
  3478     qed
  3479   qed
  3480 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
  3481 
  3482 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
  3483 proof-
  3484   { assume ?rhs
  3485     { fix e::real
  3486       assume "e>0"
  3487       with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
  3488         by (erule_tac x="e/2" in allE) auto
  3489       { fix n m
  3490         assume nm:"N \<le> m \<and> N \<le> n"
  3491         hence "dist (s m) (s n) < e" using N
  3492           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
  3493           by blast
  3494       }
  3495       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
  3496         by blast
  3497     }
  3498     hence ?lhs
  3499       unfolding cauchy_def
  3500       by blast
  3501   }
  3502   thus ?thesis
  3503     unfolding cauchy_def
  3504     using dist_triangle_half_l
  3505     by blast
  3506 qed
  3507 
  3508 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
  3509 proof-
  3510   from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto
  3511   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
  3512   moreover
  3513   have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
  3514   then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
  3515     unfolding bounded_any_center [where a="s N"] by auto
  3516   ultimately show "?thesis"
  3517     unfolding bounded_any_center [where a="s N"]
  3518     apply(rule_tac x="max a 1" in exI) apply auto
  3519     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
  3520 qed
  3521 
  3522 instance heine_borel < complete_space
  3523 proof
  3524   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  3525   hence "bounded (range f)"
  3526     by (rule cauchy_imp_bounded)
  3527   hence "compact (closure (range f))"
  3528     unfolding compact_eq_bounded_closed by auto
  3529   hence "complete (closure (range f))"
  3530     by (rule compact_imp_complete)
  3531   moreover have "\<forall>n. f n \<in> closure (range f)"
  3532     using closure_subset [of "range f"] by auto
  3533   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
  3534     using `Cauchy f` unfolding complete_def by auto
  3535   then show "convergent f"
  3536     unfolding convergent_def by auto
  3537 qed
  3538 
  3539 instance euclidean_space \<subseteq> banach ..
  3540 
  3541 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
  3542 proof(simp add: complete_def, rule, rule)
  3543   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
  3544   hence "convergent f" by (rule Cauchy_convergent)
  3545   thus "\<exists>l. f ----> l" unfolding convergent_def .  
  3546 qed
  3547 
  3548 lemma complete_imp_closed: assumes "complete s" shows "closed s"
  3549 proof -
  3550   { fix x assume "x islimpt s"
  3551     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
  3552       unfolding islimpt_sequential by auto
  3553     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
  3554       using `complete s`[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto
  3555     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
  3556   }
  3557   thus "closed s" unfolding closed_limpt by auto
  3558 qed
  3559 
  3560 lemma complete_eq_closed:
  3561   fixes s :: "'a::complete_space set"
  3562   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
  3563 proof
  3564   assume ?lhs thus ?rhs by (rule complete_imp_closed)
  3565 next
  3566   assume ?rhs
  3567   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
  3568     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
  3569     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `?rhs`[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto  }
  3570   thus ?lhs unfolding complete_def by auto
  3571 qed
  3572 
  3573 lemma convergent_eq_cauchy:
  3574   fixes s :: "nat \<Rightarrow> 'a::complete_space"
  3575   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"
  3576   unfolding Cauchy_convergent_iff convergent_def ..
  3577 
  3578 lemma convergent_imp_bounded:
  3579   fixes s :: "nat \<Rightarrow> 'a::metric_space"
  3580   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"
  3581   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
  3582 
  3583 lemma compact_cball[simp]:
  3584   fixes x :: "'a::heine_borel"
  3585   shows "compact(cball x e)"
  3586   using compact_eq_bounded_closed bounded_cball closed_cball
  3587   by blast
  3588 
  3589 lemma compact_frontier_bounded[intro]:
  3590   fixes s :: "'a::heine_borel set"
  3591   shows "bounded s ==> compact(frontier s)"
  3592   unfolding frontier_def
  3593   using compact_eq_bounded_closed
  3594   by blast
  3595 
  3596 lemma compact_frontier[intro]:
  3597   fixes s :: "'a::heine_borel set"
  3598   shows "compact s ==> compact (frontier s)"
  3599   using compact_eq_bounded_closed compact_frontier_bounded
  3600   by blast
  3601 
  3602 lemma frontier_subset_compact:
  3603   fixes s :: "'a::heine_borel set"
  3604   shows "compact s ==> frontier s \<subseteq> s"
  3605   using frontier_subset_closed compact_eq_bounded_closed
  3606   by blast
  3607 
  3608 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}
  3609 
  3610 lemma bounded_closed_nest:
  3611   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
  3612   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"
  3613   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
  3614 proof-
  3615   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto
  3616   from assms(4,1) have *:"seq_compact (s 0)" using bounded_closed_imp_seq_compact[of "s 0"] by auto
  3617 
  3618   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
  3619     unfolding seq_compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast
  3620 
  3621   { fix n::nat
  3622     { fix e::real assume "e>0"
  3623       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto
  3624       hence "dist ((x \<circ> r) (max N n)) l < e" by auto
  3625       moreover
  3626       have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"] by auto
  3627       hence "(x \<circ> r) (max N n) \<in> s n"
  3628         using x apply(erule_tac x=n in allE)
  3629         using x apply(erule_tac x="r (max N n)" in allE)
  3630         using assms(3) apply(erule_tac x=n in allE) apply(erule_tac x="r (max N n)" in allE) by auto
  3631       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
  3632     }
  3633     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
  3634   }
  3635   thus ?thesis by auto
  3636 qed
  3637 
  3638 text {* Decreasing case does not even need compactness, just completeness. *}
  3639 
  3640 lemma decreasing_closed_nest:
  3641   assumes "\<forall>n. closed(s n)"
  3642           "\<forall>n. (s n \<noteq> {})"
  3643           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3644           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
  3645   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"
  3646 proof-
  3647   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
  3648   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
  3649   then obtain t where t: "\<forall>n. t n \<in> s n" by auto
  3650   { fix e::real assume "e>0"
  3651     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
  3652     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
  3653       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+
  3654       hence "dist (t m) (t n) < e" using N by auto
  3655     }
  3656     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
  3657   }
  3658   hence  "Cauchy t" unfolding cauchy_def by auto
  3659   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
  3660   { fix n::nat
  3661     { fix e::real assume "e>0"
  3662       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto
  3663       have "t (max n N) \<in> s n" using assms(3) unfolding subset_eq apply(erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t by auto
  3664       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
  3665     }
  3666     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
  3667   }
  3668   then show ?thesis by auto
  3669 qed
  3670 
  3671 text {* Strengthen it to the intersection actually being a singleton. *}
  3672 
  3673 lemma decreasing_closed_nest_sing:
  3674   fixes s :: "nat \<Rightarrow> 'a::complete_space set"
  3675   assumes "\<forall>n. closed(s n)"
  3676           "\<forall>n. s n \<noteq> {}"
  3677           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
  3678           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
  3679   shows "\<exists>a. \<Inter>(range s) = {a}"
  3680 proof-
  3681   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
  3682   { fix b assume b:"b \<in> \<Inter>(range s)"
  3683     { fix e::real assume "e>0"
  3684       hence "dist a b < e" using assms(4 )using b using a by blast
  3685     }
  3686     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
  3687   }
  3688   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
  3689   thus ?thesis ..
  3690 qed
  3691 
  3692 text{* Cauchy-type criteria for uniform convergence. *}
  3693 
  3694 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space" shows
  3695  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
  3696   (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")
  3697 proof(rule)
  3698   assume ?lhs
  3699   then obtain l where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e" by auto
  3700   { fix e::real assume "e>0"
  3701     then obtain N::nat where N:"\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto
  3702     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
  3703       hence "dist (s m x) (s n x) < e"
  3704         using N[THEN spec[where x=m], THEN spec[where x=x]]
  3705         using N[THEN spec[where x=n], THEN spec[where x=x]]
  3706         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }
  3707     hence "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e"  by auto  }
  3708   thus ?rhs by auto
  3709 next
  3710   assume ?rhs
  3711   hence "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto
  3712   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
  3713     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
  3714   { fix e::real assume "e>0"
  3715     then obtain N where N:"\<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e/2"
  3716       using `?rhs`[THEN spec[where x="e/2"]] by auto
  3717     { fix x assume "P x"
  3718       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
  3719         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using `e>0` by(auto elim!: allE[where x="e/2"])
  3720       fix n::nat assume "n\<ge>N"
  3721       hence "dist(s n x)(l x) < e"  using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]
  3722         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute)  }
  3723     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
  3724   thus ?lhs by auto
  3725 qed
  3726 
  3727 lemma uniformly_cauchy_imp_uniformly_convergent:
  3728   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"
  3729   assumes "\<forall>e>0.\<exists>N. \<forall>m (n::nat) x. N \<le> m \<and> N \<le> n \<and> P x --> dist(s m x)(s n x) < e"
  3730           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
  3731   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
  3732 proof-
  3733   obtain l' where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l' x) < e"
  3734     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
  3735   moreover
  3736   { fix x assume "P x"
  3737     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
  3738       using l and assms(2) unfolding LIMSEQ_def by blast  }
  3739   ultimately show ?thesis by auto
  3740 qed
  3741 
  3742 
  3743 subsection {* Continuity *}
  3744 
  3745 text {* Define continuity over a net to take in restrictions of the set. *}
  3746 
  3747 definition
  3748   continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
  3749   where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
  3750 
  3751 lemma continuous_trivial_limit:
  3752  "trivial_limit net ==> continuous net f"
  3753   unfolding continuous_def tendsto_def trivial_limit_eq by auto
  3754 
  3755 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
  3756   unfolding continuous_def
  3757   unfolding tendsto_def
  3758   using netlimit_within[of x s]
  3759   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
  3760 
  3761 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
  3762   using continuous_within [of x UNIV f] by simp
  3763 
  3764 lemma continuous_isCont: "isCont f x = continuous (at x) f"
  3765   unfolding isCont_def LIM_def
  3766   unfolding continuous_at Lim_at unfolding dist_nz by auto
  3767 
  3768 lemma continuous_at_within:
  3769   assumes "continuous (at x) f"  shows "continuous (at x within s) f"
  3770   using assms unfolding continuous_at continuous_within
  3771   by (rule Lim_at_within)
  3772 
  3773 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
  3774 
  3775 lemma continuous_within_eps_delta:
  3776   "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s.  dist x' x < d --> dist (f x') (f x) < e)"
  3777   unfolding continuous_within and Lim_within
  3778   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto
  3779 
  3780 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.
  3781                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
  3782   using continuous_within_eps_delta [of x UNIV f] by simp
  3783 
  3784 text{* Versions in terms of open balls. *}
  3785 
  3786 lemma continuous_within_ball:
  3787  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  3788                             f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3789 proof
  3790   assume ?lhs
  3791   { fix e::real assume "e>0"
  3792     then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
  3793       using `?lhs`[unfolded continuous_within Lim_within] by auto
  3794     { fix y assume "y\<in>f ` (ball x d \<inter> s)"
  3795       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
  3796         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
  3797     }
  3798     hence "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e" using `d>0` unfolding subset_eq ball_def by (auto simp add: dist_commute)  }
  3799   thus ?rhs by auto
  3800 next
  3801   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
  3802     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
  3803 qed
  3804 
  3805 lemma continuous_at_ball:
  3806   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
  3807 proof
  3808   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3809     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz)
  3810     unfolding dist_nz[THEN sym] by auto
  3811 next
  3812   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
  3813     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x="f xa" in allE) by (auto simp add: dist_commute dist_nz)
  3814 qed
  3815 
  3816 text{* Define setwise continuity in terms of limits within the set. *}
  3817 
  3818 definition
  3819   continuous_on ::
  3820     "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
  3821 where
  3822   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
  3823 
  3824 lemma continuous_on_topological:
  3825   "continuous_on s f \<longleftrightarrow>
  3826     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
  3827       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
  3828 unfolding continuous_on_def tendsto_def
  3829 unfolding Limits.eventually_within eventually_at_topological
  3830 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
  3831 
  3832 lemma continuous_on_iff:
  3833   "continuous_on s f \<longleftrightarrow>
  3834     (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
  3835 unfolding continuous_on_def Lim_within
  3836 apply (intro ball_cong [OF refl] all_cong ex_cong)
  3837 apply (rename_tac y, case_tac "y = x", simp)
  3838 apply (simp add: dist_nz)
  3839 done
  3840 
  3841 definition
  3842   uniformly_continuous_on ::
  3843     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
  3844 where
  3845   "uniformly_continuous_on s f \<longleftrightarrow>
  3846     (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
  3847 
  3848 text{* Some simple consequential lemmas. *}
  3849 
  3850 lemma uniformly_continuous_imp_continuous:
  3851  " uniformly_continuous_on s f ==> continuous_on s f"
  3852   unfolding uniformly_continuous_on_def continuous_on_iff by blast
  3853 
  3854 lemma continuous_at_imp_continuous_within:
  3855  "continuous (at x) f ==> continuous (at x within s) f"
  3856   unfolding continuous_within continuous_at using Lim_at_within by auto
  3857 
  3858 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
  3859 unfolding tendsto_def by (simp add: trivial_limit_eq)
  3860 
  3861 lemma continuous_at_imp_continuous_on:
  3862   assumes "\<forall>x\<in>s. continuous (at x) f"
  3863   shows "continuous_on s f"
  3864 unfolding continuous_on_def
  3865 proof
  3866   fix x assume "x \<in> s"
  3867   with assms have *: "(f ---> f (netlimit (at x))) (at x)"
  3868     unfolding continuous_def by simp
  3869   have "(f ---> f x) (at x)"
  3870   proof (cases "trivial_limit (at x)")
  3871     case True thus ?thesis
  3872       by (rule Lim_trivial_limit)
  3873   next
  3874     case False
  3875     hence 1: "netlimit (at x) = x"
  3876       using netlimit_within [of x UNIV] by simp
  3877     with * show ?thesis by simp
  3878   qed
  3879   thus "(f ---> f x) (at x within s)"
  3880     by (rule Lim_at_within)
  3881 qed
  3882 
  3883 lemma continuous_on_eq_continuous_within:
  3884   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
  3885 unfolding continuous_on_def continuous_def
  3886 apply (rule ball_cong [OF refl])
  3887 apply (case_tac "trivial_limit (at x within s)")
  3888 apply (simp add: Lim_trivial_limit)
  3889 apply (simp add: netlimit_within)
  3890 done
  3891 
  3892 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
  3893 
  3894 lemma continuous_on_eq_continuous_at:
  3895   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
  3896   by (auto simp add: continuous_on continuous_at Lim_within_open)
  3897 
  3898 lemma continuous_within_subset:
  3899  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
  3900              ==> continuous (at x within t) f"
  3901   unfolding continuous_within by(metis Lim_within_subset)
  3902 
  3903 lemma continuous_on_subset:
  3904   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
  3905   unfolding continuous_on by (metis subset_eq Lim_within_subset)
  3906 
  3907 lemma continuous_on_interior:
  3908   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
  3909   by (erule interiorE, drule (1) continuous_on_subset,
  3910     simp add: continuous_on_eq_continuous_at)
  3911 
  3912 lemma continuous_on_eq:
  3913   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
  3914   unfolding continuous_on_def tendsto_def Limits.eventually_within
  3915   by simp
  3916 
  3917 text {* Characterization of various kinds of continuity in terms of sequences. *}
  3918 
  3919 lemma continuous_within_sequentially:
  3920   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3921   shows "continuous (at a within s) f \<longleftrightarrow>
  3922                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
  3923                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
  3924 proof
  3925   assume ?lhs
  3926   { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"
  3927     fix T::"'b set" assume "open T" and "f a \<in> T"
  3928     with `?lhs` obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> f x \<in> T"
  3929       unfolding continuous_within tendsto_def eventually_within by auto
  3930     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"
  3931       using x(2) `d>0` by simp
  3932     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"
  3933     proof eventually_elim
  3934       case (elim n) thus ?case
  3935         using d x(1) `f a \<in> T` unfolding dist_nz[THEN sym] by auto
  3936     qed
  3937   }
  3938   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp
  3939 next
  3940   assume ?rhs thus ?lhs
  3941     unfolding continuous_within tendsto_def [where l="f a"]
  3942     by (simp add: sequentially_imp_eventually_within)
  3943 qed
  3944 
  3945 lemma continuous_at_sequentially:
  3946   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3947   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
  3948                   --> ((f o x) ---> f a) sequentially)"
  3949   using continuous_within_sequentially[of a UNIV f] by simp
  3950 
  3951 lemma continuous_on_sequentially:
  3952   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  3953   shows "continuous_on s f \<longleftrightarrow>
  3954     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
  3955                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
  3956 proof
  3957   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
  3958 next
  3959   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
  3960 qed
  3961 
  3962 lemma uniformly_continuous_on_sequentially:
  3963   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
  3964                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
  3965                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
  3966 proof
  3967   assume ?lhs
  3968   { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
  3969     { fix e::real assume "e>0"
  3970       then obtain d where "d>0" and d:"\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
  3971         using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
  3972       obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded LIMSEQ_def dist_norm] and `d>0` by auto
  3973       { fix n assume "n\<ge>N"
  3974         hence "dist (f (x n)) (f (y n)) < e"
  3975           using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y
  3976           unfolding dist_commute by simp  }
  3977       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }
  3978     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }
  3979   thus ?rhs by auto
  3980 next
  3981   assume ?rhs
  3982   { assume "\<not> ?lhs"
  3983     then obtain e where "e>0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto
  3984     then obtain fa where fa:"\<forall>x.  0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
  3985       using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"] unfolding Bex_def
  3986       by (auto simp add: dist_commute)
  3987     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
  3988     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
  3989     have xyn:"\<forall>n. x n \<in> s \<and> y n \<in> s" and xy0:"\<forall>n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
  3990       unfolding x_def and y_def using fa by auto
  3991     { fix e::real assume "e>0"
  3992       then obtain N::nat where "N \<noteq> 0" and N:"0 < inverse (real N) \<and> inverse (real N) < e" unfolding real_arch_inv[of e]   by auto
  3993       { fix n::nat assume "n\<ge>N"
  3994         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
  3995         also have "\<dots> < e" using N by auto
  3996         finally have "inverse (real n + 1) < e" by auto
  3997         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }
  3998       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }
  3999     hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding LIMSEQ_def dist_real_def by auto
  4000     hence False using fxy and `e>0` by auto  }
  4001   thus ?lhs unfolding uniformly_continuous_on_def by blast
  4002 qed
  4003 
  4004 text{* The usual transformation theorems. *}
  4005 
  4006 lemma continuous_transform_within:
  4007   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  4008   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
  4009           "continuous (at x within s) f"
  4010   shows "continuous (at x within s) g"
  4011 unfolding continuous_within
  4012 proof (rule Lim_transform_within)
  4013   show "0 < d" by fact
  4014   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
  4015     using assms(3) by auto
  4016   have "f x = g x"
  4017     using assms(1,2,3) by auto
  4018   thus "(f ---> g x) (at x within s)"
  4019     using assms(4) unfolding continuous_within by simp
  4020 qed
  4021 
  4022 lemma continuous_transform_at:
  4023   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
  4024   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
  4025           "continuous (at x) f"
  4026   shows "continuous (at x) g"
  4027   using continuous_transform_within [of d x UNIV f g] assms by simp
  4028 
  4029 subsubsection {* Structural rules for pointwise continuity *}
  4030 
  4031 lemma continuous_within_id: "continuous (at a within s) (\<lambda>x. x)"
  4032   unfolding continuous_within by (rule tendsto_ident_at_within)
  4033 
  4034 lemma continuous_at_id: "continuous (at a) (\<lambda>x. x)"
  4035   unfolding continuous_at by (rule tendsto_ident_at)
  4036 
  4037 lemma continuous_const: "continuous F (\<lambda>x. c)"
  4038   unfolding continuous_def by (rule tendsto_const)
  4039 
  4040 lemma continuous_fst: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. fst (f x))"
  4041   unfolding continuous_def by (rule tendsto_fst)
  4042 
  4043 lemma continuous_snd: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. snd (f x))"
  4044   unfolding continuous_def by (rule tendsto_snd)
  4045 
  4046 lemma continuous_Pair: "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))"
  4047   unfolding continuous_def by (rule tendsto_Pair)
  4048 
  4049 lemma continuous_dist:
  4050   assumes "continuous F f" and "continuous F g"
  4051   shows "continuous F (\<lambda>x. dist (f x) (g x))"
  4052   using assms unfolding continuous_def by (rule tendsto_dist)
  4053 
  4054 lemma continuous_infdist:
  4055   assumes "continuous F f"
  4056   shows "continuous F (\<lambda>x. infdist (f x) A)"
  4057   using assms unfolding continuous_def by (rule tendsto_infdist)
  4058 
  4059 lemma continuous_norm:
  4060   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
  4061   unfolding continuous_def by (rule tendsto_norm)
  4062 
  4063 lemma continuous_infnorm:
  4064   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"
  4065   unfolding continuous_def by (rule tendsto_infnorm)
  4066 
  4067 lemma continuous_add:
  4068   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  4069   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
  4070   unfolding continuous_def by (rule tendsto_add)
  4071 
  4072 lemma continuous_minus:
  4073   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  4074   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
  4075   unfolding continuous_def by (rule tendsto_minus)
  4076 
  4077 lemma continuous_diff:
  4078   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  4079   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
  4080   unfolding continuous_def by (rule tendsto_diff)
  4081 
  4082 lemma continuous_scaleR:
  4083   fixes g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  4084   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x *\<^sub>R g x)"
  4085   unfolding continuous_def by (rule tendsto_scaleR)
  4086 
  4087 lemma continuous_mult:
  4088   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
  4089   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x * g x)"
  4090   unfolding continuous_def by (rule tendsto_mult)
  4091 
  4092 lemma continuous_inner:
  4093   assumes "continuous F f" and "continuous F g"
  4094   shows "continuous F (\<lambda>x. inner (f x) (g x))"
  4095   using assms unfolding continuous_def by (rule tendsto_inner)
  4096 
  4097 lemma continuous_inverse:
  4098   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  4099   assumes "continuous F f" and "f (netlimit F) \<noteq> 0"
  4100   shows "continuous F (\<lambda>x. inverse (f x))"
  4101   using assms unfolding continuous_def by (rule tendsto_inverse)
  4102 
  4103 lemma continuous_at_within_inverse:
  4104   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  4105   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
  4106   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
  4107   using assms unfolding continuous_within by (rule tendsto_inverse)
  4108 
  4109 lemma continuous_at_inverse:
  4110   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  4111   assumes "continuous (at a) f" and "f a \<noteq> 0"
  4112   shows "continuous (at a) (\<lambda>x. inverse (f x))"
  4113   using assms unfolding continuous_at by (rule tendsto_inverse)
  4114 
  4115 lemmas continuous_intros = continuous_at_id continuous_within_id
  4116   continuous_const continuous_dist continuous_norm continuous_infnorm
  4117   continuous_add continuous_minus continuous_diff continuous_scaleR continuous_mult
  4118   continuous_inner continuous_at_inverse continuous_at_within_inverse
  4119 
  4120 subsubsection {* Structural rules for setwise continuity *}
  4121 
  4122 lemma continuous_on_id: "continuous_on s (\<lambda>x. x)"
  4123   unfolding continuous_on_def by (fast intro: tendsto_ident_at_within)
  4124 
  4125 lemma continuous_on_const: "continuous_on s (\<lambda>x. c)"
  4126   unfolding continuous_on_def by (auto intro: tendsto_intros)
  4127 
  4128 lemma continuous_on_norm:
  4129   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
  4130   unfolding continuous_on_def by (fast intro: tendsto_norm)
  4131 
  4132 lemma continuous_on_infnorm:
  4133   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"
  4134   unfolding continuous_on by (fast intro: tendsto_infnorm)
  4135 
  4136 lemma continuous_on_minus:
  4137   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  4138   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
  4139   unfolding continuous_on_def by (auto intro: tendsto_intros)
  4140 
  4141 lemma continuous_on_add:
  4142   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  4143   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  4144            \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
  4145   unfolding continuous_on_def by (auto intro: tendsto_intros)
  4146 
  4147 lemma continuous_on_diff:
  4148   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  4149   shows "continuous_on s f \<Longrightarrow> continuous_on s g
  4150            \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
  4151   unfolding continuous_on_def by (auto intro: tendsto_intros)
  4152 
  4153 lemma (in bounded_linear) continuous_on:
  4154   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
  4155   unfolding continuous_on_def by (fast intro: tendsto)
  4156 
  4157 lemma (in bounded_bilinear) continuous_on:
  4158   "\<lbrakk>continuous_on s f; continuous_on s g\<rbrakk> \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
  4159   unfolding continuous_on_def by (fast intro: tendsto)
  4160 
  4161 lemma continuous_on_scaleR:
  4162   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  4163   assumes "continuous_on s f" and "continuous_on s g"
  4164   shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"
  4165   using bounded_bilinear_scaleR assms
  4166   by (rule bounded_bilinear.continuous_on)
  4167 
  4168 lemma continuous_on_mult:
  4169   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
  4170   assumes "continuous_on s f" and "continuous_on s g"
  4171   shows "continuous_on s (\<lambda>x. f x * g x)"
  4172   using bounded_bilinear_mult assms
  4173   by (rule bounded_bilinear.continuous_on)
  4174 
  4175 lemma continuous_on_inner:
  4176   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"
  4177   assumes "continuous_on s f" and "continuous_on s g"
  4178   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"
  4179   using bounded_bilinear_inner assms
  4180   by (rule bounded_bilinear.continuous_on)
  4181 
  4182 lemma continuous_on_inverse:
  4183   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
  4184   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
  4185   shows "continuous_on s (\<lambda>x. inverse (f x))"
  4186   using assms unfolding continuous_on by (fast intro: tendsto_inverse)
  4187 
  4188 subsubsection {* Structural rules for uniform continuity *}
  4189 
  4190 lemma uniformly_continuous_on_id:
  4191   shows "uniformly_continuous_on s (\<lambda>x. x)"
  4192   unfolding uniformly_continuous_on_def by auto
  4193 
  4194 lemma uniformly_continuous_on_const:
  4195   shows "uniformly_continuous_on s (\<lambda>x. c)"
  4196   unfolding uniformly_continuous_on_def by simp
  4197 
  4198 lemma uniformly_continuous_on_dist:
  4199   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  4200   assumes "uniformly_continuous_on s f"
  4201   assumes "uniformly_continuous_on s g"
  4202   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
  4203 proof -
  4204   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
  4205       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
  4206       using dist_triangle3 [of c d a] dist_triangle [of a d b]
  4207       by arith
  4208   } note le = this
  4209   { fix x y
  4210     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"
  4211     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"
  4212     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"
  4213       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
  4214         simp add: le)
  4215   }
  4216   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially
  4217     unfolding dist_real_def by simp
  4218 qed
  4219 
  4220 lemma uniformly_continuous_on_norm:
  4221   assumes "uniformly_continuous_on s f"
  4222   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
  4223   unfolding norm_conv_dist using assms
  4224   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
  4225 
  4226 lemma (in bounded_linear) uniformly_continuous_on:
  4227   assumes "uniformly_continuous_on s g"
  4228   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
  4229   using assms unfolding uniformly_continuous_on_sequentially
  4230   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
  4231   by (auto intro: tendsto_zero)
  4232 
  4233 lemma uniformly_continuous_on_cmul:
  4234   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4235   assumes "uniformly_continuous_on s f"
  4236   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
  4237   using bounded_linear_scaleR_right assms
  4238   by (rule bounded_linear.uniformly_continuous_on)
  4239 
  4240 lemma dist_minus:
  4241   fixes x y :: "'a::real_normed_vector"
  4242   shows "dist (- x) (- y) = dist x y"
  4243   unfolding dist_norm minus_diff_minus norm_minus_cancel ..
  4244 
  4245 lemma uniformly_continuous_on_minus:
  4246   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4247   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
  4248   unfolding uniformly_continuous_on_def dist_minus .
  4249 
  4250 lemma uniformly_continuous_on_add:
  4251   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4252   assumes "uniformly_continuous_on s f"
  4253   assumes "uniformly_continuous_on s g"
  4254   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
  4255   using assms unfolding uniformly_continuous_on_sequentially
  4256   unfolding dist_norm tendsto_norm_zero_iff add_diff_add
  4257   by (auto intro: tendsto_add_zero)
  4258 
  4259 lemma uniformly_continuous_on_diff:
  4260   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4261   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"
  4262   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
  4263   unfolding ab_diff_minus using assms
  4264   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)
  4265 
  4266 text{* Continuity of all kinds is preserved under composition. *}
  4267 
  4268 lemma continuous_within_topological:
  4269   "continuous (at x within s) f \<longleftrightarrow>
  4270     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
  4271       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
  4272 unfolding continuous_within
  4273 unfolding tendsto_def Limits.eventually_within eventually_at_topological
  4274 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
  4275 
  4276 lemma continuous_within_compose:
  4277   assumes "continuous (at x within s) f"
  4278   assumes "continuous (at (f x) within f ` s) g"
  4279   shows "continuous (at x within s) (g o f)"
  4280 using assms unfolding continuous_within_topological by simp metis
  4281 
  4282 lemma continuous_at_compose:
  4283   assumes "continuous (at x) f" and "continuous (at (f x)) g"
  4284   shows "continuous (at x) (g o f)"
  4285 proof-
  4286   have "continuous (at (f x) within range f) g" using assms(2)
  4287     using continuous_within_subset[of "f x" UNIV g "range f"] by simp
  4288   thus ?thesis using assms(1)
  4289     using continuous_within_compose[of x UNIV f g] by simp
  4290 qed
  4291 
  4292 lemma continuous_on_compose:
  4293   "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
  4294   unfolding continuous_on_topological by simp metis
  4295 
  4296 lemma uniformly_continuous_on_compose:
  4297   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f ` s) g"
  4298   shows "uniformly_continuous_on s (g o f)"
  4299 proof-
  4300   { fix e::real assume "e>0"
  4301     then obtain d where "d>0" and d:"\<forall>x\<in>f ` s. \<forall>x'\<in>f ` s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto
  4302     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d" using `d>0` using assms(1) unfolding uniformly_continuous_on_def by auto
  4303     hence "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist ((g \<circ> f) x') ((g \<circ> f) x) < e" using `d>0` using d by auto  }
  4304   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
  4305 qed
  4306 
  4307 lemmas continuous_on_intros = continuous_on_id continuous_on_const
  4308   continuous_on_compose continuous_on_norm continuous_on_infnorm
  4309   continuous_on_add continuous_on_minus continuous_on_diff
  4310   continuous_on_scaleR continuous_on_mult continuous_on_inverse
  4311   continuous_on_inner
  4312   uniformly_continuous_on_id uniformly_continuous_on_const
  4313   uniformly_continuous_on_dist uniformly_continuous_on_norm
  4314   uniformly_continuous_on_compose uniformly_continuous_on_add
  4315   uniformly_continuous_on_minus uniformly_continuous_on_diff
  4316   uniformly_continuous_on_cmul
  4317 
  4318 text{* Continuity in terms of open preimages. *}
  4319 
  4320 lemma continuous_at_open:
  4321   shows "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
  4322 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
  4323 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
  4324 
  4325 lemma continuous_on_open:
  4326   shows "continuous_on s f \<longleftrightarrow>
  4327         (\<forall>t. openin (subtopology euclidean (f ` s)) t
  4328             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
  4329 proof (safe)
  4330   fix t :: "'b set"
  4331   assume 1: "continuous_on s f"
  4332   assume 2: "openin (subtopology euclidean (f ` s)) t"
  4333   from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
  4334     unfolding openin_open by auto
  4335   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
  4336   have "open U" unfolding U_def by (simp add: open_Union)
  4337   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
  4338   proof (intro ballI iffI)
  4339     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
  4340       unfolding U_def t by auto
  4341   next
  4342     fix x assume "x \<in> s" and "f x \<in> t"
  4343     hence "x \<in> s" and "f x \<in> B"
  4344       unfolding t by auto
  4345     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
  4346       unfolding t continuous_on_topological by metis
  4347     then show "x \<in> U"
  4348       unfolding U_def by auto
  4349   qed
  4350   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
  4351   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  4352     unfolding openin_open by fast
  4353 next
  4354   assume "?rhs" show "continuous_on s f"
  4355   unfolding continuous_on_topological
  4356   proof (clarify)
  4357     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
  4358     have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
  4359       unfolding openin_open using `open B` by auto
  4360     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
  4361       using `?rhs` by fast
  4362     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
  4363       unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
  4364   qed
  4365 qed
  4366 
  4367 text {* Similarly in terms of closed sets. *}
  4368 
  4369 lemma continuous_on_closed:
  4370   shows "continuous_on s f \<longleftrightarrow>  (\<forall>t. closedin (subtopology euclidean (f ` s)) t  --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
  4371 proof
  4372   assume ?lhs
  4373   { fix t
  4374     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  4375     have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
  4376     assume as:"closedin (subtopology euclidean (f ` s)) t"
  4377     hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
  4378     hence "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?lhs`[unfolded continuous_on_open, THEN spec[where x="(f ` s) - t"]]
  4379       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }
  4380   thus ?rhs by auto
  4381 next
  4382   assume ?rhs
  4383   { fix t
  4384     have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
  4385     assume as:"openin (subtopology euclidean (f ` s)) t"
  4386     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
  4387       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
  4388   thus ?lhs unfolding continuous_on_open by auto
  4389 qed
  4390 
  4391 text {* Half-global and completely global cases. *}
  4392 
  4393 lemma continuous_open_in_preimage:
  4394   assumes "continuous_on s f"  "open t"
  4395   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  4396 proof-
  4397   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
  4398   have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  4399     using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
  4400   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  4401 qed
  4402 
  4403 lemma continuous_closed_in_preimage:
  4404   assumes "continuous_on s f"  "closed t"
  4405   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
  4406 proof-
  4407   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
  4408   have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
  4409     using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
  4410   thus ?thesis
  4411     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
  4412 qed
  4413 
  4414 lemma continuous_open_preimage:
  4415   assumes "continuous_on s f" "open s" "open t"
  4416   shows "open {x \<in> s. f x \<in> t}"
  4417 proof-
  4418   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  4419     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
  4420   thus ?thesis using open_Int[of s T, OF assms(2)] by auto
  4421 qed
  4422 
  4423 lemma continuous_closed_preimage:
  4424   assumes "continuous_on s f" "closed s" "closed t"
  4425   shows "closed {x \<in> s. f x \<in> t}"
  4426 proof-
  4427   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
  4428     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
  4429   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
  4430 qed
  4431 
  4432 lemma continuous_open_preimage_univ:
  4433   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
  4434   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
  4435 
  4436 lemma continuous_closed_preimage_univ:
  4437   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
  4438   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
  4439 
  4440 lemma continuous_open_vimage:
  4441   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
  4442   unfolding vimage_def by (rule continuous_open_preimage_univ)
  4443 
  4444 lemma continuous_closed_vimage:
  4445   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
  4446   unfolding vimage_def by (rule continuous_closed_preimage_univ)
  4447 
  4448 lemma interior_image_subset:
  4449   assumes "\<forall>x. continuous (at x) f" "inj f"
  4450   shows "interior (f ` s) \<subseteq> f ` (interior s)"
  4451 proof
  4452   fix x assume "x \<in> interior (f ` s)"
  4453   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` s" ..
  4454   hence "x \<in> f ` s" by auto
  4455   then obtain y where y: "y \<in> s" "x = f y" by auto
  4456   have "open (vimage f T)"
  4457     using assms(1) `open T` by (rule continuous_open_vimage)
  4458   moreover have "y \<in> vimage f T"
  4459     using `x = f y` `x \<in> T` by simp
  4460   moreover have "vimage f T \<subseteq> s"
  4461     using `T \<subseteq> image f s` `inj f` unfolding inj_on_def subset_eq by auto
  4462   ultimately have "y \<in> interior s" ..
  4463   with `x = f y` show "x \<in> f ` interior s" ..
  4464 qed
  4465 
  4466 text {* Equality of continuous functions on closure and related results. *}
  4467 
  4468 lemma continuous_closed_in_preimage_constant:
  4469   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4470   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
  4471   using continuous_closed_in_preimage[of s f "{a}"] by auto
  4472 
  4473 lemma continuous_closed_preimage_constant:
  4474   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4475   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
  4476   using continuous_closed_preimage[of s f "{a}"] by auto
  4477 
  4478 lemma continuous_constant_on_closure:
  4479   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4480   assumes "continuous_on (closure s) f"
  4481           "\<forall>x \<in> s. f x = a"
  4482   shows "\<forall>x \<in> (closure s). f x = a"
  4483     using continuous_closed_preimage_constant[of "closure s" f a]
  4484     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
  4485 
  4486 lemma image_closure_subset:
  4487   assumes "continuous_on (closure s) f"  "closed t"  "(f ` s) \<subseteq> t"
  4488   shows "f ` (closure s) \<subseteq> t"
  4489 proof-
  4490   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
  4491   moreover have "closed {x \<in> closure s. f x \<in> t}"
  4492     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
  4493   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
  4494     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
  4495   thus ?thesis by auto
  4496 qed
  4497 
  4498 lemma continuous_on_closure_norm_le:
  4499   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  4500   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"
  4501   shows "norm(f x) \<le> b"
  4502 proof-
  4503   have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
  4504   show ?thesis
  4505     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
  4506     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
  4507 qed
  4508 
  4509 text {* Making a continuous function avoid some value in a neighbourhood. *}
  4510 
  4511 lemma continuous_within_avoid:
  4512   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
  4513   assumes "continuous (at x within s) f" and "f x \<noteq> a"
  4514   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
  4515 proof-
  4516   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"
  4517     using t1_space [OF `f x \<noteq> a`] by fast
  4518   have "(f ---> f x) (at x within s)"
  4519     using assms(1) by (simp add: continuous_within)
  4520   hence "eventually (\<lambda>y. f y \<in> U) (at x within s)"
  4521     using `open U` and `f x \<in> U`
  4522     unfolding tendsto_def by fast
  4523   hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"
  4524     using `a \<notin> U` by (fast elim: eventually_mono [rotated])
  4525   thus ?thesis
  4526     unfolding Limits.eventually_within Limits.eventually_at
  4527     by (rule ex_forward, cut_tac `f x \<noteq> a`, auto simp: dist_commute)
  4528 qed
  4529 
  4530 lemma continuous_at_avoid:
  4531   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
  4532   assumes "continuous (at x) f" and "f x \<noteq> a"
  4533   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  4534   using assms continuous_within_avoid[of x UNIV f a] by simp
  4535 
  4536 lemma continuous_on_avoid:
  4537   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
  4538   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"
  4539   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
  4540 using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)]  continuous_within_avoid[of x s f a]  assms(3) by auto
  4541 
  4542 lemma continuous_on_open_avoid:
  4543   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
  4544   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"
  4545   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
  4546 using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]  continuous_at_avoid[of x f a]  assms(4) by auto
  4547 
  4548 text {* Proving a function is constant by proving open-ness of level set. *}
  4549 
  4550 lemma continuous_levelset_open_in_cases:
  4551   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4552   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  4553         openin (subtopology euclidean s) {x \<in> s. f x = a}
  4554         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
  4555 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
  4556 
  4557 lemma continuous_levelset_open_in:
  4558   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4559   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
  4560         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
  4561         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"
  4562 using continuous_levelset_open_in_cases[of s f ]
  4563 by meson
  4564 
  4565 lemma continuous_levelset_open:
  4566   fixes f :: "_ \<Rightarrow> 'b::t1_space"
  4567   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"
  4568   shows "\<forall>x \<in> s. f x = a"
  4569 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
  4570 
  4571 text {* Some arithmetical combinations (more to prove). *}
  4572 
  4573 lemma open_scaling[intro]:
  4574   fixes s :: "'a::real_normed_vector set"
  4575   assumes "c \<noteq> 0"  "open s"
  4576   shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
  4577 proof-
  4578   { fix x assume "x \<in> s"
  4579     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto
  4580     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
  4581     moreover
  4582     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
  4583       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
  4584         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
  4585           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
  4586       hence "y \<in> op *\<^sub>R c ` s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]  e[THEN spec[where x="(1 / c) *\<^sub>R y"]]  assms(1) unfolding dist_norm scaleR_scaleR by auto  }
  4587     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s" apply(rule_tac x="e * abs c" in exI) by auto  }
  4588   thus ?thesis unfolding open_dist by auto
  4589 qed
  4590 
  4591 lemma minus_image_eq_vimage:
  4592   fixes A :: "'a::ab_group_add set"
  4593   shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
  4594   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
  4595 
  4596 lemma open_negations:
  4597   fixes s :: "'a::real_normed_vector set"
  4598   shows "open s ==> open ((\<lambda> x. -x) ` s)"
  4599   unfolding scaleR_minus1_left [symmetric]
  4600   by (rule open_scaling, auto)
  4601 
  4602 lemma open_translation:
  4603   fixes s :: "'a::real_normed_vector set"
  4604   assumes "open s"  shows "open((\<lambda>x. a + x) ` s)"
  4605 proof-
  4606   { fix x have "continuous (at x) (\<lambda>x. x - a)"
  4607       by (intro continuous_diff continuous_at_id continuous_const) }
  4608   moreover have "{x. x - a \<in> s} = op + a ` s" by force
  4609   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
  4610 qed
  4611 
  4612 lemma open_affinity:
  4613   fixes s :: "'a::real_normed_vector set"
  4614   assumes "open s"  "c \<noteq> 0"
  4615   shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  4616 proof-
  4617   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
  4618   have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
  4619   thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
  4620 qed
  4621 
  4622 lemma interior_translation:
  4623   fixes s :: "'a::real_normed_vector set"
  4624   shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
  4625 proof (rule set_eqI, rule)
  4626   fix x assume "x \<in> interior (op + a ` s)"
  4627   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
  4628   hence "ball (x - a) e \<subseteq> s" unfolding subset_eq Ball_def mem_ball dist_norm apply auto apply(erule_tac x="a + xa" in allE) unfolding ab_group_add_class.diff_diff_eq[THEN sym] by auto
  4629   thus "x \<in> op + a ` interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using `e > 0` by auto
  4630 next
  4631   fix x assume "x \<in> op + a ` interior s"
  4632   then obtain y e where "e>0" and e:"ball y e \<subseteq> s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto
  4633   { fix z have *:"a + y - z = y + a - z" by auto
  4634     assume "z\<in>ball x e"
  4635     hence "z - a \<in> s" using e[unfolded subset_eq, THEN bspec[where x="z - a"]] unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 * by auto
  4636     hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }
  4637   hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
  4638   thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
  4639 qed
  4640 
  4641 text {* Topological properties of linear functions. *}
  4642 
  4643 lemma linear_lim_0:
  4644   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
  4645 proof-
  4646   interpret f: bounded_linear f by fact
  4647   have "(f ---> f 0) (at 0)"
  4648     using tendsto_ident_at by (rule f.tendsto)
  4649   thus ?thesis unfolding f.zero .
  4650 qed
  4651 
  4652 lemma linear_continuous_at:
  4653   assumes "bounded_linear f"  shows "continuous (at a) f"
  4654   unfolding continuous_at using assms
  4655   apply (rule bounded_linear.tendsto)
  4656   apply (rule tendsto_ident_at)
  4657   done
  4658 
  4659 lemma linear_continuous_within:
  4660   shows "bounded_linear f ==> continuous (at x within s) f"
  4661   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
  4662 
  4663 lemma linear_continuous_on:
  4664   shows "bounded_linear f ==> continuous_on s f"
  4665   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
  4666 
  4667 text {* Also bilinear functions, in composition form. *}
  4668 
  4669 lemma bilinear_continuous_at_compose:
  4670   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
  4671         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
  4672   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
  4673 
  4674 lemma bilinear_continuous_within_compose:
  4675   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
  4676         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
  4677   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
  4678 
  4679 lemma bilinear_continuous_on_compose:
  4680   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
  4681              ==> continuous_on s (\<lambda>x. h (f x) (g x))"
  4682   unfolding continuous_on_def
  4683   by (fast elim: bounded_bilinear.tendsto)
  4684 
  4685 text {* Preservation of compactness and connectedness under continuous function. *}
  4686 
  4687 lemma compact_eq_openin_cover:
  4688   "compact S \<longleftrightarrow>
  4689     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
  4690       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
  4691 proof safe
  4692   fix C
  4693   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"
  4694   hence "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"
  4695     unfolding openin_open by force+
  4696   with `compact S` obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"
  4697     by (rule compactE)
  4698   hence "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"
  4699     by auto
  4700   thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
  4701 next
  4702   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
  4703         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"
  4704   show "compact S"
  4705   proof (rule compactI)
  4706     fix C
  4707     let ?C = "image (\<lambda>T. S \<inter> T) C"
  4708     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"
  4709     hence "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"
  4710       unfolding openin_open by auto
  4711     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"
  4712       by metis
  4713     let ?D = "inv_into C (\<lambda>T. S \<inter> T) ` D"
  4714     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"
  4715     proof (intro conjI)
  4716       from `D \<subseteq> ?C` show "?D \<subseteq> C"
  4717         by (fast intro: inv_into_into)
  4718       from `finite D` show "finite ?D"
  4719         by (rule finite_imageI)
  4720       from `S \<subseteq> \<Union>D` show "S \<subseteq> \<Union>?D"
  4721         apply (rule subset_trans)
  4722         apply clarsimp
  4723         apply (frule subsetD [OF `D \<subseteq> ?C`, THEN f_inv_into_f])
  4724         apply (erule rev_bexI, fast)
  4725         done
  4726     qed
  4727     thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
  4728   qed
  4729 qed
  4730 
  4731 lemma compact_continuous_image:
  4732   assumes "continuous_on s f" and "compact s"
  4733   shows "compact (f ` s)"
  4734 using assms (* FIXME: long unstructured proof *)
  4735 unfolding continuous_on_open
  4736 unfolding compact_eq_openin_cover
  4737 apply clarify
  4738 apply (drule_tac x="image (\<lambda>t. {x \<in> s. f x \<in> t}) C" in spec)
  4739 apply (drule mp)
  4740 apply (rule conjI)
  4741 apply simp
  4742 apply clarsimp
  4743 apply (drule subsetD)
  4744 apply (erule imageI)
  4745 apply fast
  4746 apply (erule thin_rl)
  4747 apply clarify
  4748 apply (rule_tac x="image (inv_into C (\<lambda>t. {x \<in> s. f x \<in> t})) D" in exI)
  4749 apply (intro conjI)
  4750 apply clarify
  4751 apply (rule inv_into_into)
  4752 apply (erule (1) subsetD)
  4753 apply (erule finite_imageI)
  4754 apply (clarsimp, rename_tac x)
  4755 apply (drule (1) subsetD, clarify)
  4756 apply (drule (1) subsetD, clarify)
  4757 apply (rule rev_bexI)
  4758 apply assumption
  4759 apply (subgoal_tac "{x \<in> s. f x \<in> t} \<in> (\<lambda>t. {x \<in> s. f x \<in> t}) ` C")
  4760 apply (drule f_inv_into_f)
  4761 apply fast
  4762 apply (erule imageI)
  4763 done
  4764 
  4765 lemma connected_continuous_image:
  4766   assumes "continuous_on s f"  "connected s"
  4767   shows "connected(f ` s)"
  4768 proof-
  4769   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f ` s"  "openin (subtopology euclidean (f ` s)) T"  "closedin (subtopology euclidean (f ` s)) T"
  4770     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
  4771       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
  4772       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
  4773       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
  4774     hence False using as(1,2)
  4775       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
  4776   thus ?thesis unfolding connected_clopen by auto
  4777 qed
  4778 
  4779 text {* Continuity implies uniform continuity on a compact domain. *}
  4780   
  4781 lemma compact_uniformly_continuous:
  4782   assumes f: "continuous_on s f" and s: "compact s"
  4783   shows "uniformly_continuous_on s f"
  4784   unfolding uniformly_continuous_on_def
  4785 proof (cases, safe)
  4786   fix e :: real assume "0 < e" "s \<noteq> {}"
  4787   def [simp]: R \<equiv> "{(y, d). y \<in> s \<and> 0 < d \<and> ball y d \<inter> s \<subseteq> {x \<in> s. f x \<in> ball (f y) (e/2) } }"
  4788   let ?b = "(\<lambda>(y, d). ball y (d/2))"
  4789   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"
  4790   proof safe
  4791     fix y assume "y \<in> s"
  4792     from continuous_open_in_preimage[OF f open_ball]
  4793     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"
  4794       unfolding openin_subtopology open_openin by metis
  4795     then obtain d where "ball y d \<subseteq> T" "0 < d"
  4796       using `0 < e` `y \<in> s` by (auto elim!: openE)
  4797     with T `y \<in> s` show "y \<in> (\<Union>r\<in>R. ?b r)"
  4798       by (intro UN_I[of "(y, d)"]) auto
  4799   qed auto
  4800   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"
  4801     by (rule compactE_image)
  4802   with `s \<noteq> {}` have [simp]: "\<And>x. x < Min (snd ` D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"
  4803     by (subst Min_gr_iff) auto
  4804   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
  4805   proof (rule, safe)
  4806     fix x x' assume in_s: "x' \<in> s" "x \<in> s"
  4807     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"
  4808       by blast
  4809     moreover assume "dist x x' < Min (snd`D) / 2"
  4810     ultimately have "dist y x' < d"
  4811       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)
  4812     with D x in_s show  "dist (f x) (f x') < e"
  4813       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)
  4814   qed (insert D, auto)
  4815 qed auto
  4816 
  4817 text{* Continuity of inverse function on compact domain. *}
  4818 
  4819 lemma continuous_on_inv:
  4820   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
  4821   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"
  4822   shows "continuous_on (f ` s) g"
  4823 unfolding continuous_on_topological
  4824 proof (clarsimp simp add: assms(3))
  4825   fix x :: 'a and B :: "'a set"
  4826   assume "x \<in> s" and "open B" and "x \<in> B"
  4827   have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
  4828     using assms(3) by (auto, metis)
  4829   have "continuous_on (s - B) f"
  4830     using `continuous_on s f` Diff_subset
  4831     by (rule continuous_on_subset)
  4832   moreover have "compact (s - B)"
  4833     using `open B` and `compact s`
  4834     unfolding Diff_eq by (intro compact_inter_closed closed_Compl)
  4835   ultimately have "compact (f ` (s - B))"
  4836     by (rule compact_continuous_image)
  4837   hence "closed (f ` (s - B))"
  4838     by (rule compact_imp_closed)
  4839   hence "open (- f ` (s - B))"
  4840     by (rule open_Compl)
  4841   moreover have "f x \<in> - f ` (s - B)"
  4842     using `x \<in> s` and `x \<in> B` by (simp add: 1)
  4843   moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
  4844     by (simp add: 1)
  4845   ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"
  4846     by fast
  4847 qed
  4848 
  4849 text {* A uniformly convergent limit of continuous functions is continuous. *}
  4850 
  4851 lemma continuous_uniform_limit:
  4852   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
  4853   assumes "\<not> trivial_limit F"
  4854   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
  4855   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
  4856   shows "continuous_on s g"
  4857 proof-
  4858   { fix x and e::real assume "x\<in>s" "e>0"
  4859     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
  4860       using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
  4861     from eventually_happens [OF eventually_conj [OF this assms(2)]]
  4862     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"
  4863       using assms(1) by blast
  4864     have "e / 3 > 0" using `e>0` by auto
  4865     then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
  4866       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
  4867     { fix y assume "y \<in> s" and "dist y x < d"
  4868       hence "dist (f n y) (f n x) < e / 3"
  4869         by (rule d [rule_format])
  4870       hence "dist (f n y) (g x) < 2 * e / 3"
  4871         using dist_triangle [of "f n y" "g x" "f n x"]
  4872         using n(1)[THEN bspec[where x=x], OF `x\<in>s`]
  4873         by auto
  4874       hence "dist (g y) (g x) < e"
  4875         using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
  4876         using dist_triangle3 [of "g y" "g x" "f n y"]
  4877         by auto }
  4878     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
  4879       using `d>0` by auto }
  4880   thus ?thesis unfolding continuous_on_iff by auto
  4881 qed
  4882 
  4883 
  4884 subsection {* Topological stuff lifted from and dropped to R *}
  4885 
  4886 lemma open_real:
  4887   fixes s :: "real set" shows
  4888  "open s \<longleftrightarrow>
  4889         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
  4890   unfolding open_dist dist_norm by simp
  4891 
  4892 lemma islimpt_approachable_real:
  4893   fixes s :: "real set"
  4894   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
  4895   unfolding islimpt_approachable dist_norm by simp
  4896 
  4897 lemma closed_real:
  4898   fixes s :: "real set"
  4899   shows "closed s \<longleftrightarrow>
  4900         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
  4901             --> x \<in> s)"
  4902   unfolding closed_limpt islimpt_approachable dist_norm by simp
  4903 
  4904 lemma continuous_at_real_range:
  4905   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4906   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
  4907         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
  4908   unfolding continuous_at unfolding Lim_at
  4909   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
  4910   apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto
  4911   apply(erule_tac x=e in allE) by auto
  4912 
  4913 lemma continuous_on_real_range:
  4914   fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  4915   shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
  4916   unfolding continuous_on_iff dist_norm by simp
  4917 
  4918 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}
  4919 
  4920 lemma compact_attains_sup:
  4921   fixes s :: "real set"
  4922   assumes "compact s"  "s \<noteq> {}"
  4923   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
  4924 proof-
  4925   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
  4926   { fix e::real assume as: "\<forall>x\<in>s. x \<le> Sup s" "Sup s \<notin> s"  "0 < e" "\<forall>x'\<in>s. x' = Sup s \<or> \<not> Sup s - x' < e"
  4927     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
  4928     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
  4929     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto  }
  4930   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
  4931     apply(rule_tac x="Sup s" in bexI) by auto
  4932 qed
  4933 
  4934 lemma Inf:
  4935   fixes S :: "real set"
  4936   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
  4937 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def) 
  4938 
  4939 lemma compact_attains_inf:
  4940   fixes s :: "real set"
  4941   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
  4942 proof-
  4943   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
  4944   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"
  4945       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
  4946     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
  4947     moreover
  4948     { fix x assume "x \<in> s"
  4949       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
  4950       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
  4951     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
  4952     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto  }
  4953   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
  4954     apply(rule_tac x="Inf s" in bexI) by auto
  4955 qed
  4956 
  4957 lemma continuous_attains_sup:
  4958   fixes f :: "'a::topological_space \<Rightarrow> real"
  4959   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
  4960         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"
  4961   using compact_attains_sup[of "f ` s"]
  4962   using compact_continuous_image[of s f] by auto
  4963 
  4964 lemma continuous_attains_inf:
  4965   fixes f :: "'a::topological_space \<Rightarrow> real"
  4966   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
  4967         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
  4968   using compact_attains_inf[of "f ` s"]
  4969   using compact_continuous_image[of s f] by auto
  4970 
  4971 lemma distance_attains_sup:
  4972   assumes "compact s" "s \<noteq> {}"
  4973   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
  4974 proof (rule continuous_attains_sup [OF assms])
  4975   { fix x assume "x\<in>s"
  4976     have "(dist a ---> dist a x) (at x within s)"
  4977       by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)
  4978   }
  4979   thus "continuous_on s (dist a)"
  4980     unfolding continuous_on ..
  4981 qed
  4982 
  4983 text {* For \emph{minimal} distance, we only need closure, not compactness. *}
  4984 
  4985 lemma distance_attains_inf:
  4986   fixes a :: "'a::heine_borel"
  4987   assumes "closed s"  "s \<noteq> {}"
  4988   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
  4989 proof-
  4990   from assms(2) obtain b where "b\<in>s" by auto
  4991   let ?B = "cball a (dist b a) \<inter> s"
  4992   have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
  4993   hence "?B \<noteq> {}" by auto
  4994   moreover
  4995   { fix x assume "x\<in>?B"
  4996     fix e::real assume "e>0"
  4997     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
  4998       from as have "\<bar>dist a x' - dist a x\<bar> < e"
  4999         unfolding abs_less_iff minus_diff_eq
  5000         using dist_triangle2 [of a x' x]
  5001         using dist_triangle [of a x x']
  5002         by arith
  5003     }
  5004     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
  5005       using `e>0` by auto
  5006   }
  5007   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
  5008     unfolding continuous_on Lim_within dist_norm real_norm_def
  5009     by fast
  5010   moreover have "compact ?B"
  5011     using compact_cball[of a "dist b a"]
  5012     unfolding compact_eq_bounded_closed
  5013     using bounded_Int and closed_Int and assms(1) by auto
  5014   ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y"
  5015     using continuous_attains_inf[of ?B "dist a"] by fastforce
  5016   thus ?thesis by fastforce
  5017 qed
  5018 
  5019 
  5020 subsection {* Pasted sets *}
  5021 
  5022 lemma bounded_Times:
  5023   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
  5024 proof-
  5025   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
  5026     using assms [unfolded bounded_def] by auto
  5027   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
  5028     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
  5029   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
  5030 qed
  5031 
  5032 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
  5033 by (induct x) simp
  5034 
  5035 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
  5036 unfolding seq_compact_def
  5037 apply clarify
  5038 apply (drule_tac x="fst \<circ> f" in spec)
  5039 apply (drule mp, simp add: mem_Times_iff)
  5040 apply (clarify, rename_tac l1 r1)
  5041 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
  5042 apply (drule mp, simp add: mem_Times_iff)
  5043 apply (clarify, rename_tac l2 r2)
  5044 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
  5045 apply (rule_tac x="r1 \<circ> r2" in exI)
  5046 apply (rule conjI, simp add: subseq_def)
  5047 apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
  5048 apply (drule (1) tendsto_Pair) back
  5049 apply (simp add: o_def)
  5050 done
  5051 
  5052 text {* Generalize to @{class topological_space} *}
  5053 lemma compact_Times: 
  5054   fixes s :: "'a::metric_space set" and t :: "'b::metric_space set"
  5055   shows "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
  5056   unfolding compact_eq_seq_compact_metric by (rule seq_compact_Times)
  5057 
  5058 text{* Hence some useful properties follow quite easily. *}
  5059 
  5060 lemma compact_scaling:
  5061   fixes s :: "'a::real_normed_vector set"
  5062   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
  5063 proof-
  5064   let ?f = "\<lambda>x. scaleR c x"
  5065   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)
  5066   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
  5067     using linear_continuous_at[OF *] assms by auto
  5068 qed
  5069 
  5070 lemma compact_negations:
  5071   fixes s :: "'a::real_normed_vector set"
  5072   assumes "compact s"  shows "compact ((\<lambda>x. -x) ` s)"
  5073   using compact_scaling [OF assms, of "- 1"] by auto
  5074 
  5075 lemma compact_sums:
  5076   fixes s t :: "'a::real_normed_vector set"
  5077   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
  5078 proof-
  5079   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
  5080     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
  5081   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
  5082     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  5083   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
  5084 qed
  5085 
  5086 lemma compact_differences:
  5087   fixes s t :: "'a::real_normed_vector set"
  5088   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
  5089 proof-
  5090   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
  5091     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  5092   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
  5093 qed
  5094 
  5095 lemma compact_translation:
  5096   fixes s :: "'a::real_normed_vector set"
  5097   assumes "compact s"  shows "compact ((\<lambda>x. a + x) ` s)"
  5098 proof-
  5099   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
  5100   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
  5101 qed
  5102 
  5103 lemma compact_affinity:
  5104   fixes s :: "'a::real_normed_vector set"
  5105   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
  5106 proof-
  5107   have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
  5108   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
  5109 qed
  5110 
  5111 text {* Hence we get the following. *}
  5112 
  5113 lemma compact_sup_maxdistance:
  5114   fixes s :: "'a::metric_space set"
  5115   assumes "compact s"  "s \<noteq> {}"
  5116   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
  5117 proof-
  5118   have "compact (s \<times> s)" using `compact s` by (intro compact_Times)
  5119   moreover have "s \<times> s \<noteq> {}" using `s \<noteq> {}` by auto
  5120   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"
  5121     by (intro continuous_at_imp_continuous_on ballI continuous_dist
  5122       continuous_isCont[THEN iffD1] isCont_fst isCont_snd isCont_ident)
  5123   ultimately show ?thesis
  5124     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto
  5125 qed
  5126 
  5127 text {* We can state this in terms of diameter of a set. *}
  5128 
  5129 definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"
  5130 
  5131 lemma diameter_bounded_bound:
  5132   fixes s :: "'a :: metric_space set"
  5133   assumes s: "bounded s" "x \<in> s" "y \<in> s"
  5134   shows "dist x y \<le> diameter s"
  5135 proof -
  5136   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
  5137   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"
  5138     unfolding bounded_def by auto
  5139   have "dist x y \<le> Sup ?D"
  5140   proof (rule Sup_upper, safe)
  5141     fix a b assume "a \<in> s" "b \<in> s"
  5142     with z[of a] z[of b] dist_triangle[of a b z]
  5143     show "dist a b \<le> 2 * d"
  5144       by (simp add: dist_commute)
  5145   qed (insert s, auto)
  5146   with `x \<in> s` show ?thesis
  5147     by (auto simp add: diameter_def)
  5148 qed
  5149 
  5150 lemma diameter_lower_bounded:
  5151   fixes s :: "'a :: metric_space set"
  5152   assumes s: "bounded s" and d: "0 < d" "d < diameter s"
  5153   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"
  5154 proof (rule ccontr)
  5155   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
  5156   assume contr: "\<not> ?thesis"
  5157   moreover
  5158   from d have "s \<noteq> {}"
  5159     by (auto simp: diameter_def)
  5160   then have "?D \<noteq> {}" by auto
  5161   ultimately have "Sup ?D \<le> d"
  5162     by (intro Sup_least) (auto simp: not_less)
  5163   with `d < diameter s` `s \<noteq> {}` show False
  5164     by (auto simp: diameter_def)
  5165 qed
  5166 
  5167 lemma diameter_bounded:
  5168   assumes "bounded s"
  5169   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"
  5170         "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"
  5171   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms
  5172   by auto
  5173 
  5174 lemma diameter_compact_attained:
  5175   assumes "compact s"  "s \<noteq> {}"
  5176   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"
  5177 proof -
  5178   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
  5179   then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
  5180     using compact_sup_maxdistance[OF assms] by auto
  5181   hence "diameter s \<le> dist x y"
  5182     unfolding diameter_def by clarsimp (rule Sup_least, fast+)
  5183   thus ?thesis
  5184     by (metis b diameter_bounded_bound order_antisym xys)
  5185 qed
  5186 
  5187 text {* Related results with closure as the conclusion. *}
  5188 
  5189 lemma closed_scaling:
  5190   fixes s :: "'a::real_normed_vector set"
  5191   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
  5192 proof(cases "s={}")
  5193   case True thus ?thesis by auto
  5194 next
  5195   case False
  5196   show ?thesis
  5197   proof(cases "c=0")
  5198     have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
  5199     case True thus ?thesis apply auto unfolding * by auto
  5200   next
  5201     case False
  5202     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s"  "(x ---> l) sequentially"
  5203       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
  5204           using as(1)[THEN spec[where x=n]]
  5205           using `c\<noteq>0` by auto
  5206       }
  5207       moreover
  5208       { fix e::real assume "e>0"
  5209         hence "0 < e *\<bar>c\<bar>"  using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
  5210         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
  5211           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto
  5212         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
  5213           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
  5214           using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto  }
  5215       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto
  5216       ultimately have "l \<in> scaleR c ` s"
  5217         using assms[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]]
  5218         unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }
  5219     thus ?thesis unfolding closed_sequential_limits by fast
  5220   qed
  5221 qed
  5222 
  5223 lemma closed_negations:
  5224   fixes s :: "'a::real_normed_vector set"
  5225   assumes "closed s"  shows "closed ((\<lambda>x. -x) ` s)"
  5226   using closed_scaling[OF assms, of "- 1"] by simp
  5227 
  5228 lemma compact_closed_sums:
  5229   fixes s :: "'a::real_normed_vector set"
  5230   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  5231 proof-
  5232   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
  5233   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"
  5234     from as(1) obtain f where f:"\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> s"  "\<forall>n. snd (f n) \<in> t"
  5235       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
  5236     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
  5237       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
  5238     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
  5239       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1) unfolding o_def by auto
  5240     hence "l - l' \<in> t"
  5241       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
  5242       using f(3) by auto
  5243     hence "l \<in> ?S" using `l' \<in> s` apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto
  5244   }
  5245   thus ?thesis unfolding closed_sequential_limits by fast
  5246 qed
  5247 
  5248 lemma closed_compact_sums:
  5249   fixes s t :: "'a::real_normed_vector set"
  5250   assumes "closed s"  "compact t"
  5251   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
  5252 proof-
  5253   have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}" apply auto
  5254     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
  5255   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
  5256 qed
  5257 
  5258 lemma compact_closed_differences:
  5259   fixes s t :: "'a::real_normed_vector set"
  5260   assumes "compact s"  "closed t"
  5261   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  5262 proof-
  5263   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"
  5264     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  5265   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
  5266 qed
  5267 
  5268 lemma closed_compact_differences:
  5269   fixes s t :: "'a::real_normed_vector set"
  5270   assumes "closed s" "compact t"
  5271   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
  5272 proof-
  5273   have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
  5274     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
  5275  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
  5276 qed
  5277 
  5278 lemma closed_translation:
  5279   fixes a :: "'a::real_normed_vector"
  5280   assumes "closed s"  shows "closed ((\<lambda>x. a + x) ` s)"
  5281 proof-
  5282   have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
  5283   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
  5284 qed
  5285 
  5286 lemma translation_Compl:
  5287   fixes a :: "'a::ab_group_add"
  5288   shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
  5289   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
  5290 
  5291 lemma translation_UNIV:
  5292   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
  5293   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
  5294 
  5295 lemma translation_diff:
  5296   fixes a :: "'a::ab_group_add"
  5297   shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
  5298   by auto
  5299 
  5300 lemma closure_translation:
  5301   fixes a :: "'a::real_normed_vector"
  5302   shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
  5303 proof-
  5304   have *:"op + a ` (- s) = - op + a ` s"
  5305     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
  5306   show ?thesis unfolding closure_interior translation_Compl
  5307     using interior_translation[of a "- s"] unfolding * by auto
  5308 qed
  5309 
  5310 lemma frontier_translation:
  5311   fixes a :: "'a::real_normed_vector"
  5312   shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
  5313   unfolding frontier_def translation_diff interior_translation closure_translation by auto
  5314 
  5315 
  5316 subsection {* Separation between points and sets *}
  5317 
  5318 lemma separate_point_closed:
  5319   fixes s :: "'a::heine_borel set"
  5320   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
  5321 proof(cases "s = {}")
  5322   case True
  5323   thus ?thesis by(auto intro!: exI[where x=1])
  5324 next
  5325   case False
  5326   assume "closed s" "a \<notin> s"
  5327   then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using `s \<noteq> {}` distance_attains_inf [of s a] by blast
  5328   with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
  5329 qed
  5330 
  5331 lemma separate_compact_closed:
  5332   fixes s t :: "'a::heine_borel set"
  5333   assumes "compact s" and "closed t" and "s \<inter> t = {}"
  5334   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  5335 proof - (* FIXME: long proof *)
  5336   let ?T = "\<Union>x\<in>s. { ball x (d / 2) | d. 0 < d \<and> (\<forall>y\<in>t. d \<le> dist x y) }"
  5337   note `compact s`
  5338   moreover have "\<forall>t\<in>?T. open t" by auto
  5339   moreover have "s \<subseteq> \<Union>?T"
  5340     apply auto
  5341     apply (rule rev_bexI, assumption)
  5342     apply (subgoal_tac "x \<notin> t")
  5343     apply (drule separate_point_closed [OF `closed t`])
  5344     apply clarify
  5345     apply (rule_tac x="ball x (d / 2)" in exI)
  5346     apply simp
  5347     apply fast
  5348     apply (cut_tac assms(3))
  5349     apply auto
  5350     done
  5351   ultimately obtain U where "U \<subseteq> ?T" and "finite U" and "s \<subseteq> \<Union>U"
  5352     by (rule compactE)
  5353   from `finite U` and `U \<subseteq> ?T` have "\<exists>d>0. \<forall>x\<in>\<Union>U. \<forall>y\<in>t. d \<le> dist x y"
  5354     apply (induct set: finite)
  5355     apply simp
  5356     apply (rule exI)
  5357     apply (rule zero_less_one)
  5358     apply clarsimp
  5359     apply (rename_tac y e)
  5360     apply (rule_tac x="min d (e / 2)" in exI)
  5361     apply simp
  5362     apply (subst ball_Un)
  5363     apply (rule conjI)
  5364     apply (intro ballI, rename_tac z)
  5365     apply (rule min_max.le_infI2)
  5366     apply (simp only: mem_ball)
  5367     apply (drule (1) bspec)
  5368     apply (cut_tac x=y and y=x and z=z in dist_triangle, arith)
  5369     apply simp
  5370     apply (intro ballI)
  5371     apply (rule min_max.le_infI1)
  5372     apply simp
  5373     done
  5374   with `s \<subseteq> \<Union>U` show ?thesis
  5375     by fast
  5376 qed
  5377 
  5378 lemma separate_closed_compact:
  5379   fixes s t :: "'a::heine_borel set"
  5380   assumes "closed s" and "compact t" and "s \<inter> t = {}"
  5381   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
  5382 proof-
  5383   have *:"t \<inter> s = {}" using assms(3) by auto
  5384   show ?thesis using separate_compact_closed[OF assms(2,1) *]
  5385     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
  5386     by (auto simp add: dist_commute)
  5387 qed
  5388 
  5389 
  5390 subsection {* Intervals *}
  5391   
  5392 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
  5393   "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and
  5394   "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"
  5395   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
  5396 
  5397 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
  5398   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"
  5399   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"
  5400   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
  5401 
  5402 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
  5403  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and
  5404  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
  5405 proof-
  5406   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"
  5407     hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto
  5408     hence "a\<bullet>i < b\<bullet>i" by auto
  5409     hence False using as by auto  }
  5410   moreover
  5411   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
  5412     let ?x = "(1/2) *\<^sub>R (a + b)"
  5413     { fix i :: 'a assume i:"i\<in>Basis" 
  5414       have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto
  5415       hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
  5416         by (auto simp: inner_add_left) }
  5417     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
  5418   ultimately show ?th1 by blast
  5419 
  5420   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"
  5421     hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto
  5422     hence "a\<bullet>i \<le> b\<bullet>i" by auto
  5423     hence False using as by auto  }
  5424   moreover
  5425   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
  5426     let ?x = "(1/2) *\<^sub>R (a + b)"
  5427     { fix i :: 'a assume i:"i\<in>Basis"
  5428       have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto
  5429       hence "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
  5430         by (auto simp: inner_add_left) }
  5431     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
  5432   ultimately show ?th2 by blast
  5433 qed
  5434 
  5435 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
  5436   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and
  5437   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
  5438   unfolding interval_eq_empty[of a b] by fastforce+
  5439 
  5440 lemma interval_sing:
  5441   fixes a :: "'a::ordered_euclidean_space"
  5442   shows "{a .. a} = {a}" and "{a<..<a} = {}"
  5443   unfolding set_eq_iff mem_interval eq_iff [symmetric]
  5444   by (auto intro: euclidean_eqI simp: ex_in_conv)
  5445 
  5446 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
  5447  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
  5448  "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
  5449  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
  5450  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
  5451   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
  5452   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
  5453 
  5454 lemma interval_open_subset_closed:
  5455   fixes a :: "'a::ordered_euclidean_space"
  5456   shows "{a<..<b} \<subseteq> {a .. b}"
  5457   unfolding subset_eq [unfolded Ball_def] mem_interval
  5458   by (fast intro: less_imp_le)
  5459 
  5460 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
  5461  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1) and
  5462  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2) and
  5463  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3) and
  5464  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
  5465 proof-
  5466   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
  5467   show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
  5468   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
  5469     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
  5470     fix i :: 'a assume i:"i\<in>Basis"
  5471     (** TODO combine the following two parts as done in the HOL_light version. **)
  5472     { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
  5473       assume as2: "a\<bullet>i > c\<bullet>i"
  5474       { fix j :: 'a assume j:"j\<in>Basis"
  5475         hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
  5476           apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i
  5477           by (auto simp add: as2)  }
  5478       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
  5479       moreover
  5480       have "?x\<notin>{a .. b}"
  5481         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
  5482         using as(2)[THEN bspec[where x=i]] and as2 i
  5483         by auto
  5484       ultimately have False using as by auto  }
  5485     hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto
  5486     moreover
  5487     { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
  5488       assume as2: "b\<bullet>i < d\<bullet>i"
  5489       { fix j :: 'a assume "j\<in>Basis"
  5490         hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j" 
  5491           apply(cases "j=i") using as(2)[THEN bspec[where x=j]]
  5492           by (auto simp add: as2) }
  5493       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
  5494       moreover
  5495       have "?x\<notin>{a .. b}"
  5496         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
  5497         using as(2)[THEN bspec[where x=i]] and as2 using i
  5498         by auto
  5499       ultimately have False using as by auto  }
  5500     hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto
  5501     ultimately
  5502     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
  5503   } note part1 = this
  5504   show ?th3
  5505     unfolding subset_eq and Ball_def and mem_interval 
  5506     apply(rule,rule,rule,rule) 
  5507     apply(rule part1)
  5508     unfolding subset_eq and Ball_def and mem_interval
  5509     prefer 4
  5510     apply auto 
  5511     by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+ 
  5512   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
  5513     fix i :: 'a assume i:"i\<in>Basis"
  5514     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
  5515     hence "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" using part1 and as(2) using i by auto  } note * = this
  5516   show ?th4 unfolding subset_eq and Ball_def and mem_interval 
  5517     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
  5518     apply auto by(erule_tac x=xa in allE, simp)+ 
  5519 qed
  5520 
  5521 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
  5522  "{a .. b} \<inter> {c .. d} =  {(\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)}"
  5523   unfolding set_eq_iff and Int_iff and mem_interval by auto
  5524 
  5525 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
  5526   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1) and
  5527   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2) and
  5528   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3) and
  5529   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
  5530 proof-
  5531   let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"
  5532   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
  5533       (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)" 
  5534     by blast
  5535   note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)
  5536   show ?th1 unfolding * by (intro **) auto
  5537   show ?th2 unfolding * by (intro **) auto
  5538   show ?th3 unfolding * by (intro **) auto
  5539   show ?th4 unfolding * by (intro **) auto
  5540 qed
  5541 
  5542 (* Moved interval_open_subset_closed a bit upwards *)
  5543 
  5544 lemma open_interval[intro]:
  5545   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
  5546 proof-
  5547   have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i<..<b\<bullet>i})"
  5548     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
  5549       linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)
  5550   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i<..<b\<bullet>i}) = {a<..<b}"
  5551     by (auto simp add: eucl_less [where 'a='a])
  5552   finally show "open {a<..<b}" .
  5553 qed
  5554 
  5555 lemma closed_interval[intro]:
  5556   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
  5557 proof-
  5558   have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i})"
  5559     by (intro closed_INT ballI continuous_closed_vimage allI
  5560       linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
  5561   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) -` {a\<bullet>i .. b\<bullet>i}) = {a .. b}"
  5562     by (auto simp add: eucl_le [where 'a='a])
  5563   finally show "closed {a .. b}" .
  5564 qed
  5565 
  5566 lemma interior_closed_interval [intro]:
  5567   fixes a b :: "'a::ordered_euclidean_space"
  5568   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")
  5569 proof(rule subset_antisym)
  5570   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval
  5571     by (rule interior_maximal)
  5572 next
  5573   { fix x assume "x \<in> interior {a..b}"
  5574     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..
  5575     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
  5576     { fix i :: 'a assume i:"i\<in>Basis"
  5577       have "dist (x - (e / 2) *\<^sub>R i) x < e"
  5578            "dist (x + (e / 2) *\<^sub>R i) x < e"
  5579         unfolding dist_norm apply auto
  5580         unfolding norm_minus_cancel using norm_Basis[OF i] `e>0` by auto
  5581       hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"
  5582                      "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"
  5583         using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]
  5584         and   e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]
  5585         unfolding mem_interval using i by blast+
  5586       hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"
  5587         using `e>0` i by (auto simp: inner_diff_left inner_Basis inner_add_left) }
  5588     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }
  5589   thus "?L \<subseteq> ?R" ..
  5590 qed
  5591 
  5592 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
  5593 proof-
  5594   let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"
  5595   { fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
  5596     { fix i :: 'a assume "i\<in>Basis"
  5597       hence "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>" using x[THEN bspec[where x=i]] by auto  }
  5598     hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
  5599     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }
  5600   thus ?thesis unfolding interval and bounded_iff by auto
  5601 qed
  5602 
  5603 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
  5604  "bounded {a .. b} \<and> bounded {a<..<b}"
  5605   using bounded_closed_interval[of a b]
  5606   using interval_open_subset_closed[of a b]
  5607   using bounded_subset[of "{a..b}" "{a<..<b}"]
  5608   by simp
  5609 
  5610 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
  5611  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
  5612   using bounded_interval[of a b] by auto
  5613 
  5614 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
  5615   using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]
  5616   by (auto simp: compact_eq_seq_compact_metric)
  5617 
  5618 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
  5619   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
  5620 proof-
  5621   { fix i :: 'a assume "i\<in>Basis"
  5622     hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"
  5623       using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }
  5624   thus ?thesis unfolding mem_interval by auto
  5625 qed
  5626 
  5627 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
  5628   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
  5629   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
  5630 proof-
  5631   { fix i :: 'a assume i:"i\<in>Basis"
  5632     have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp
  5633     also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)
  5634       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  5635       using x unfolding mem_interval using i apply simp
  5636       using y unfolding mem_interval using i apply simp
  5637       done
  5638     finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto
  5639     moreover {
  5640     have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp
  5641     also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)
  5642       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
  5643       using x unfolding mem_interval using i apply simp
  5644       using y unfolding mem_interval using i apply simp
  5645       done
  5646     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" unfolding inner_simps by auto
  5647     } ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" by auto }
  5648   thus ?thesis unfolding mem_interval by auto
  5649 qed
  5650 
  5651 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
  5652   assumes "{a<..<b} \<noteq> {}"
  5653   shows "closure {a<..<b} = {a .. b}"
  5654 proof-
  5655   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
  5656   let ?c = "(1 / 2) *\<^sub>R (a + b)"
  5657   { fix x assume as:"x \<in> {a .. b}"
  5658     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
  5659     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
  5660       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
  5661       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
  5662         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
  5663         by (auto simp add: algebra_simps)
  5664       hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
  5665       hence False using fn unfolding f_def using xc by auto  }
  5666     moreover
  5667     { assume "\<not> (f ---> x) sequentially"
  5668       { fix e::real assume "e>0"
  5669         hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
  5670         then obtain N::nat where "inverse (real (N + 1)) < e" by auto
  5671         hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
  5672         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }
  5673       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
  5674         unfolding LIMSEQ_def by(auto simp add: dist_norm)
  5675       hence "(f ---> x) sequentially" unfolding f_def
  5676         using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
  5677         using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto  }
  5678     ultimately have "x \<in> closure {a<..<b}"
  5679       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }
  5680   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
  5681 qed
  5682 
  5683 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
  5684   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"
  5685 proof-
  5686   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
  5687   def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"
  5688   { fix x assume "x\<in>s"
  5689     fix i :: 'a assume i:"i\<in>Basis"
  5690     hence "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i" using b[THEN bspec[where x=x], OF `x\<in>s`]
  5691       and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto }
  5692   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
  5693 qed
  5694 
  5695 lemma bounded_subset_open_interval:
  5696   fixes s :: "('a::ordered_euclidean_space) set"
  5697   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
  5698   by (auto dest!: bounded_subset_open_interval_symmetric)
  5699 
  5700 lemma bounded_subset_closed_interval_symmetric:
  5701   fixes s :: "('a::ordered_euclidean_space) set"
  5702   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
  5703 proof-
  5704   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
  5705   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
  5706 qed
  5707 
  5708 lemma bounded_subset_closed_interval:
  5709   fixes s :: "('a::ordered_euclidean_space) set"
  5710   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
  5711   using bounded_subset_closed_interval_symmetric[of s] by auto
  5712 
  5713 lemma frontier_closed_interval:
  5714   fixes a b :: "'a::ordered_euclidean_space"
  5715   shows "frontier {a .. b} = {a .. b} - {a<..<b}"
  5716   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
  5717 
  5718 lemma frontier_open_interval:
  5719   fixes a b :: "'a::ordered_euclidean_space"
  5720   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
  5721 proof(cases "{a<..<b} = {}")
  5722   case True thus ?thesis using frontier_empty by auto
  5723 next
  5724   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
  5725 qed
  5726 
  5727 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
  5728   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
  5729   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
  5730 
  5731 
  5732 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)
  5733 
  5734 lemma closed_interval_left: fixes b::"'a::euclidean_space"
  5735   shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"
  5736 proof-
  5737   { fix i :: 'a assume i:"i\<in>Basis"
  5738     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. x \<bullet> i \<le> b \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"
  5739     { assume "x\<bullet>i > b\<bullet>i"
  5740       then obtain y where "y \<bull